Note that the same Hebrew word for angel could either mean an angelical being, but it could also mean a messenger from God which could be literally anything like a human, bush, donkey, fire, even Jesus himself.

There are many "angels" that appear to people in the Old Testament, but to the "angel of the LORD" is different from the other ones:

Why people belief he is Jesus

The angel of the LORD acts like Jesus

And he blessed Joseph and said:

“The God before whom my fathers Abraham and Isaac walked, the God who has been my shepherd all my life long to this day, the angel who has redeemed me from all evil, bless the boys; and in them let my name be carried on, and the name of my fathers Abraham and Isaac; and let them grow into a multitude in the midst of the earth.”

-- Genesis 48:15-16 (ESV)

And the LORD said to Moses, "Thus you shall say to the people of Israel...

Behold, I send an angel before you to guard you on the way and to bring you to the place that I have prepared. Pay careful attention to him and obey his voice; do not rebel against him, for he will not pardon your transgression, for my name is in him. But if you carefully obey his voice and do all that I say, then I will be an enemy to your enemies and an adversary to your adversaries." “When my angel goes before you and brings you to the Amorites and the Hittites and the Perizzites and the Canaanites, the Hivites and the Jebusites, and I blot them out, you shall not bow down to their gods nor serve them, nor do as they do, but you shall utterly overthrow them and break their pillars in pieces. You shall serve the LORD your God, and he will bless your bread and your water, and I will take sickness away from among you."

-- Exodus 23:20-25 (ESV)

The angel of the LORD encamps around those who fear him, and delivers them. -- Psalm 34:7 (ESV)

Then he showed me Joshua the high priest standing before the angel of the LORD, and Satan standing at his right hand to accuse him.

And the LORD said to Satan, “The LORD rebuke you, O Satan! The LORD who has chosen Jerusalem rebuke you! Is not this a brand plucked from the fire?”

Now Joshua was standing before the angel, clothed with filthy garments.

And the angel said to those who were standing before him, “Remove the filthy garments from him.”
And to him he said, “Behold, I have taken your iniquity away from you, and I will clothe you with pure vestments.
And I said, “Let them put a clean turban on his head.”

So they put a clean turban on his head and clothed him with garments. And the angel of the LORD was standing by.

And the angel of the LORD solemnly assured Joshua, “Thus says the LORD of hosts:

If you will walk in My ways and keep My charge, then you shall rule My house and have charge of My courts, and I will give you the right of access among those who are standing here. Hear now, O Joshua the high priest, you and your friends who sit before you, for they are men who are a sign: behold, I will bring My servant the Branch. For behold, on the stone that I have set before Joshua, on a single stone with seven eyes, I will engrave its inscription, declares the LORD of hosts, and I will remove the iniquity of this land in a single day. In that day, declares the LORD of hosts, every one of you will invite his neighbor to come under his vine and under his fig tree.”

-- Zechariah 3:1-10 (ESV)

Assuming all of the above scriptures refers to the same "angel of the LORD" we can conclude that this angel:

• The angel is not God Himself (because God sent him).
• The angel speak and act on behalf of God.
• The angel bless generations.

If we compare "the angel of the LORD" with "the Son of God":

Although this seems to make a good case that Jesus could be "the angle of the LORD", is worth pointing out that these are not enough evidence to make a solid conclusion because:

• God also sent other humans
• God also require us to obey authority of humans
• Many prophets who are not Jesus, spoke on behalf of God
• We should also fear the human authority that God appointed above us, like kings
• Jesus also told his disciples that they too could forgive sin (John 20:23)
• The angel does not necessary have the same authority as Jesus
• "Wonderful" is not a unique attribute
• Everyone should obey God

Therefore, some assume that the "Lord's angel" is some sort of pre-incarnation of Jesus.

Jesus referring to himself as an "angel"

“Behold, I send my messenger (John the Baptist according to Matthew 11:10; Mark 1:2; Luke 7:27), and he will prepare the way before me. -- Malachi 3:1

If we assume that John prepared the way for Jesus, it would imply that Jesus is the second "messenger" or "angel".

And the Lord whom you seek will suddenly come to his temple; and the messenger (angel) of the covenant in whom you delight, behold, he is coming, says the LORD of hosts.

But who can endure the day of his coming, and who can stand when he appears?

For he is like a refiner's fire and like fullers' soap. He will sit as a refiner and purifier of silver, and he will purify the sons of Levi and refine them like gold and silver, and they will bring offerings in righteousness to the LORD. Then the offering of Judah and Jerusalem will be pleasing to the LORD as in the days of old and as in former years.

-- Malachi 3:1-4 (ESV)

Some suggest that the angel was God's instrument to fulfilled Jesus' role in his absence, so that this angel was speaking and acting on behalf of God and/or Jesus with the same authority and power:

• This is also the case with the sun, moon and stars which are temporary instruments to give light on behalf of God (Genesis 1:16-18; Matthew 24:29; Revelation 21:23,22:5)
• God can delegate his authority to whomever He wishes (Daniel 7:27; John 14:12)

Jesus considered to be the angel of John's Revelation

The Revelation of Jesus Christ, which God gave him to show His servants — things which must shortly take place. And He sent and signified it by His angel to His servant John, who bore witness to the word of God, and to the testimony of Jesus Christ, to all things that he saw. -- Revelation 1:1-2 (NKJV)

• Some interpret this that God is Jesus who gave the revelation to another angel which reveals it to John.
• Other interpret this that Jesus is the angel who is testifying to John.

Then he said to me, “These words are faithful and true.”

And the Lord God of the holy prophets sent His angel to show His servants the things which must shortly take place.

“Behold, I am coming quickly! Blessed is he who keeps the words of the prophecy of this book.”

Now I, John, saw and heard these things. And when I heard and saw, I fell down to worship before the feet of the angel who showed me these things. Then he said to me, “See that you do not do that. For I am your fellow servant, and of your brethren the prophets, and of those who keep the words of this book. Worship God.” -- Revelation 22:6-9 (NKJV)

Why would it be significant that the angel "is coming quickly", when John just revealed the Christ is returning?

In addition, the angel wields immense power throughout Revelation, opening seals, pronouncing judgments, and guiding John's visions. Jesus is also depicted with comparable authority in other parts of the New Testament.

Critiques argue, that Jesus is addressed as "the Son of God", "Lord", "the Christ" which is not the case with this angel. Another challenge is that because the book of Revelations does not follow chronologically and because there are so much symbolism, it is hard to tell to whom every "he"-word refers which makes it easy confuse God, Jesus and the angel with each other.

More examples where the "Angel of the LORD" speak and act on behalf of God

Appearance to Hagar

The angel of the LORD found her by a spring of water in the wilderness, the spring on the way to Shur.

And he said, “Hagar, servant of Sarai, where have you come from and where are you going?”
She said, “I am fleeing from my mistress Sarai.”
The angel of the LORD said to her, “Return to your mistress and submit to her.”
The angel of the LORD also said to her, “I will surely multiply your offspring so that they cannot be numbered for multitude.”
And the angel of the LORD said to her, “Behold, you are pregnant and shall bear a son. You shall call his name Ishmael, because the LORD has listened to your affliction. He shall be a wild donkey of a man, his hand against everyone and everyone's hand against him, and he shall dwell over against all his kinsmen.”

-- Genesis 16:7-12 (ESV)

The angel referred to "the LORD" as a separate person.

Appearance to Abraham

Then Abraham reached out his hand and took the knife to slaughter his son.

But the angel of the LORD called to him from heaven and said, “Abraham, Abraham!” And he said, “Here I am.” He said, “Do not lay your hand on the boy or do anything to him, for now I know that you fear God, seeing you have not withheld your son, your only son, from me.”

And Abraham lifted up his eyes and looked, and behold, behind him was a ram, caught in a thicket by his horns. And Abraham went and took the ram and offered it up as a burnt offering instead of his son.

So Abraham called the name of that place, “The LORD will provide”; as it is said to this day, “On the mount of the LORD it shall be provided.”

And the angel of the LORD called to Abraham a second time from heaven and said, “By myself I have sworn, declares the LORD, because you have done this and have not withheld your son, your only son, I will surely bless you, and I will surely multiply your offspring as the stars of heaven and as the sand that is on the seashore. And your offspring shall possess the gate of his enemies, and in your offspring shall all the nations of the earth be blessed, because you have obeyed my voice.”

-- Genesis 22:10-18 (ESV)

The angel referred to "God" as a separate God and also indicates that he "declares" on behalf of "the LORD" when he speaks.

Appearance to Moses

And the angel of the LORD appeared to him in a flame of fire out of the midst of a bush. He looked, and behold, the bush was burning, yet it was not consumed. And Moses said, “I will turn aside to see this great sight, why the bush is not burned.” When the LORD saw that he turned aside to see, God called to him out of the bush, “Moses, Moses!” And he said, “Here I am.” Then He said, “Do not come near; take your sandals off your feet, for the place on which you are standing is holy ground.” And He said, “I am the God of your father, the God of Abraham, the God of Isaac, and the God of Jacob.”

And Moses hid his face, for he was afraid to look at God.

-- Exodus 3:2-6 (ESV)

Moses saw the angel of the LORD and not God Himself (John 1:18 (NKJV); John 6:46; 1 John 3:6, 4:12; Exodus 33:20; Isaiah 64:4), however Moses did hear God's voice (perhaps God was speaking through the angel).

Jesus also confirms that it was God speaking to Moses:

Jesus said to them... "Have you not read in the book of Moses, in the passage about the bush, how God spoke to him... -- Mark 12:26 (ESV)

Jesus did not say the angel was God, or that he is the angel. He only said that "God spoke" to Moses.

Appearance to Balaam

But God's anger was kindled because he went, and the angel of the LORD took his stand in the way as his adversary. Now he was riding on the donkey, and his two servants were with him. And the donkey saw the angel of the LORD standing in the road, with a drawn sword in his hand. And the donkey turned aside out of the road and went into the field. And Balaam struck the donkey, to turn her into the road. Then the angel of the LORD stood in a narrow path between the vineyards, with a wall on either side. And when the donkey saw the angel of the LORD, she pushed against the wall and pressed Balaam's foot against the wall. So he struck her again. Then the angel of the LORD went ahead and stood in a narrow place, where there was no way to turn either to the right or to the left. When the donkey saw the angel of the LORD, she lay down under Balaam. And Balaam's anger was kindled, and he struck the donkey with his staff.

Then the LORD opened the mouth of the donkey, and she said to Balaam, “What have I done to you, that you have struck me these three times?” And Balaam said to the donkey, “Because you have made a fool of me. I wish I had a sword in my hand, for then I would kill you.” And the donkey said to Balaam, “Am I not your donkey, on which you have ridden all your life long to this day? Is it my habit to treat you this way?” And he said, “No.”

Then the LORD opened the eyes of Balaam, and he saw the angel of the LORD standing in the way, with his drawn sword in his hand. And he bowed down and fell on his face.

And the angel of the LORD said to him, “Why have you struck your donkey these three times? Behold, I have come out to oppose you because your way is perverse before me. The donkey saw me and turned aside before me these three times. If she had not turned aside from me, surely just now I would have killed you and let her live.” Then Balaam said to the angel of the LORD, “I have sinned, for I did not know that you stood in the road against me. Now therefore, if it is evil in your sight, I will turn back.” And the angel of the LORD said to Balaam, “Go with the men, but speak only the word that I tell you.” So Balaam went on with the princes of Balak.

-- Numbers 22:22-35 (ESV)

Appearance to the Israelites at Bochim

Both "the LORD" and "the angel of the LORD" are present in this scripture and did different things. The donkey and Balaam saw "the angel of the LORD", but "the LORD opened the mouth of the donkey" and their eyes.

Now the angel of the LORD went up from Gilgal to Bochim.

And he said, “I brought you up from Egypt and brought you into the land that I swore to give to your fathers. I said, I will never break my covenant with you, and you shall make no covenant with the inhabitants of this land; you shall break down their altars.’ But you have not obeyed my voice. What is this you have done? So now I say, I will not drive them out before you, but they shall become thorns in your sides, and their gods shall be a snare to you.”

As soon as the angel of the LORD spoke these words to all the people of Israel, the people lifted up their voices and wept. And they called the name of that place Bochim. And they sacrificed there to the LORD.

-- Judges 2:1-5

This implies the angel of the LORD is able to judge and make covenants. As the angel refers back to the past, it also implies that it was the same "angel of the LORD" who previously brought the Israelites out of Egypt.

Also note that they sacrificed to "the LORD" and not to "the angel".

Song of Deborah and Barak

"Curse Meroz", says the angel of the LORD, "curse its inhabitants thoroughly, because they did not come to the help of the LORD, to the help of the LORD against the mighty." -- Judges 5:23

Appearance to Gideon

Now the angel of the LORD came and sat under the terebinth at Ophrah, which belonged to Joash the Abiezrite, while his son Gideon was beating out wheat in the winepress to hide it from the Midianites.

And the angel of the LORD appeared to him and said to him, “The LORD is with you, O mighty man of valor.” And Gideon said to him, “Please, my Lord, if the LORD is with us, why then has all this happened to us? And where are all his wonderful deeds that our fathers recounted to us, saying, ‘Did not the LORD bring us up from Egypt?’ But now the LORD has forsaken us and given us into the hand of Midian.” And the LORD turned to him and said, “Go in this might of yours and save Israel from the hand of Midian; do not I send you?” And he said to him, “Please, Lord, how can I save Israel? Behold, my clan is the weakest in Manasseh, and I am the least in my father's house.” And the LORD said to him, “But I will be with you, and you shall strike the Midianites as one man.” And he said to him, “If now I have found favor in your eyes, then show me a sign that it is you who speak with me. Please do not depart from here until I come to you and bring out my present and set it before you.” And he said, “I will stay till you return.”

So Gideon went into his house and prepared a young goat and unleavened cakes from an ephah of flour. The meat he put in a basket, and the broth he put in a pot, and brought them to him under the terebinth and presented them.

And the angel of God said to him, “Take the meat and the unleavened cakes, and put them on this rock, and pour the broth over them.”

And he did so. Then the angel of the LORD reached out the tip of the staff that was in his hand and touched the meat and the unleavened cakes. And fire sprang up from the rock and consumed the meat and the unleavened cakes. And the angel of the LORD vanished from his sight. Then Gideon perceived that he was the angel of the LORD.

And Gideon said, “Alas, O Lord GOD! For now I have seen the angel of the LORD face to face.”

But the LORD said to him, “Peace be to you. Do not fear; you shall not die.”

Then Gideon built an altar there to the LORD and called it, The LORD Is Peace. To this day it still stands at Ophrah, which belongs to the Abiezrites.

That night the LORD said to him, “Take your father's bull, and the second bull seven years old, and pull down the altar of Baal that your father has, and cut down the Asherah that is beside it and build an altar to the LORD your God on the top of the stronghold here, with stones laid in due order. Then take the second bull and offer it as a burnt offering with the wood of the Asherah that you shall cut down.”

So Gideon took ten men of his servants and did as the LORD had told him. But because he was too afraid of his family and the men of the town to do it by day, he did it by night.

-- Judges 6:11-27 (ESV)

Like Moses, Gideon saw "the angel of the LORD", but heard the voice of "the LORD" even after "the angel of the LORD" had vanished from his sight.

Appearance to Samson's parents

And the angel of the LORD appeared to the woman and said to her, “Behold, you are barren and have not borne children, but you shall conceive and bear a son. Therefore be careful and drink no wine or strong drink, and eat nothing unclean, for behold, you shall conceive and bear a son. No razor shall come upon his head, for the child shall be a Nazirite to God from the womb, and he shall begin to save Israel from the hand of the Philistines.”

Then the woman came and told her husband, “A man of God came to me, and his appearance was like the appearance of the angel of God, very awesome. I did not ask him where he was from, and he did not tell me his name, but he said to me, behold, you shall conceive and bear a son. So then drink no wine or strong drink, and eat nothing unclean, for the child shall be a Nazirite to God from the womb to the day of his death.’”

Then Manoah prayed to the LORD and said, “O Lord, please let the man of God whom you sent come again to us and teach us what we are to do with the child who will be born.”

And God listened to the voice of Manoah, and the angel of God came again to the woman as she sat in the field. But Manoah her husband was not with her. So the woman ran quickly and told her husband,

“Behold, the man who came to me the other day has appeared to me.” And Manoah arose and went after his wife and came to the man and said to him, “Are you the man who spoke to this woman?” And he said, “I am.” And Manoah said, “Now when your words come true, what is to be the child's manner of life, and what is his mission?” And the angel of the LORD said to Manoah, “Of all that I said to the woman let her be careful. She may not eat of anything that comes from the vine, neither let her drink wine or strong drink, or eat any unclean thing. All that I commanded her let her observe.” Manoah said to the angel of the LORD, “Please let us detain you and prepare a young goat for you.” And the angel of the LORD said to Manoah, “If you detain me, I will not eat of your food. But if you prepare a burnt offering, then offer it to the LORD.” (For Manoah did not know that he was the angel of the LORD.) And Manoah said to the angel of the LORD, “What is your name, so that, when your words come true, we may honor you?” And the angel of the LORD said to him, “Why do you ask my name, seeing it is wonderful?”

So Manoah took the young goat with the grain offering, and offered it on the rock to the LORD, to the one who works wonders, and Manoah and his wife were watching. And when the flame went up toward heaven from the altar, the angel of the LORD went up in the flame of the altar. Now Manoah and his wife were watching, and they fell on their faces to the ground. The angel of the LORD appeared no more to Manoah and to his wife. Then Manoah knew that he was the angel of the LORD.

And Manoah said to his wife, “We shall surely die, for we have seen God.” But his wife said to him, “If the LORD had meant to kill us, he would not have accepted a burnt offering and a grain offering at our hands, or shown us all these things, or now announced to us such things as these.”

And the woman bore a son and called his name Samson. And the young man grew, and the LORD blessed him. And the Spirit of the LORD began to stir him in Mahaneh-dan, between Zorah and Eshtaol.

-- Judges 13:3-25 (ESV)

Appearance to David

So Gad came to David and said to him, “Thus says the LORD, ‘Choose what you will: either three years of famine, or three months of devastation by your foes while the sword of your enemies overtakes you, or else three days of the sword of the Lord, pestilence on the land, with the angel of the LORD destroying throughout all the territory of Israel.’ Now decide what answer I shall return to him who sent me.”
Then David said to Gad, “I am in great distress. Let me fall into the hand of the LORD, for his mercy is very great, but do not let me fall into the hand of man.”

So the LORD sent a pestilence on Israel, and 70,000 men of Israel fell. And God sent the angel to Jerusalem to destroy it, but as he was about to destroy it, the LORD saw, and He relented from the calamity.

And He said to the angel who was working destruction, “It is enough; now stay your hand.”

And the angel of the LORD was standing by the threshing floor of Ornan the Jebusite. And David lifted his eyes and saw the angel of the LORD standing between earth and heaven, and in his hand a drawn sword stretched out over Jerusalem.

-- 1 Chronicles 21:11-16 (ESV); 2 Samuel 24:13-16

Appearance to Zechariah

Then I said, ‘What are these, my lord?’ The angel who talked with me said to me, ‘I will show you what they are.’ So the man who was standing among the myrtle trees answered, ‘These are they whom the LORD has sent to patrol the earth.’ And they answered the angel of the Lord who was standing among the myrtle trees, and said, ‘We have patrolled the earth, and behold, all the earth remains at rest.’ Then the angel of the LORD said, ‘O LORD of hosts, how long will You have no mercy on Jerusalem and the cities of Judah, against which You have been angry these seventy years?’

And the LORD answered gracious and comforting words to the angel who talked with me.

So the angel who talked with me said to me, ‘Cry out, Thus says the Lord of hosts: I am exceedingly jealous for Jerusalem and for Zion. And I am exceedingly angry with the nations that are cat ease; for while I was angry but a little, they furthered the disaster. Therefore, thus says the LORD, I have returned to Jerusalem with mercy; my house shall be built in it, declares the LORD of hosts, and the measuring line shall be stretched out over Jerusalem. Cry out again, Thus says the LORD of hosts: My cities shall again overflow with prosperity, and the LORD will again comfort Zion and again choose Jerusalem.’ ”

-- Zechariah 1:9-17 (ESV)

Appearance to Joseph (Jesus' stepfather)

But as he (Joseph) considered these things, behold, an angel of the Lord appeared to him in a dream, saying, “Joseph, son of David, do not fear to take Mary as your wife, for that which is conceived in her is from the Holy Spirit. She will bear a son, and you shall call his name Jesus, for he will save his people from their sins.”

...

When Joseph woke from sleep, he did as the angel of the Lord commanded him: he took his wife.

-- Matthew 1:20-21,24 (ESV)

Appearance to the shepherds

And, lo, the angel of the Lord came upon them, and the glory of the Lord shone round about them: and they were sore afraid. -- Luke 2:9 (KJV)

Appeared the apostles

And laid their hands on the apostles, and put them in the common prison. But the angel of the Lord by night opened the prison doors, and brought them forth, and said, "Go, stand and speak in the temple to the people all the words of this life." -- Acts 5:18-20 (KJV)

And the angel of the Lord spake unto Philip, saying, "Arise, and go toward the south unto the way that goeth down from Jerusalem unto Gaza, which is desert." -- Acts 8:26 (KJV)

And, behold, the angel of the Lord came upon him, and a light shined in the prison: and he smote Peter on the side, and raised him up, saying, Arise up quickly.

And his chains fell off from his hands.

And the angel said unto him, Gird thyself, and bind on thy sandals.

And so he did.

And he saith unto him, Cast thy garment about thee, and follow me.

And he went out, and followed him; and wist not that it was true which was done by the angel; but thought he saw a vision. When they were past the first and the second ward, they came unto the iron gate that leadeth unto the city; which opened to them of his own accord: and they went out, and passed on through one street; and forthwith the angel departed from him.

And when Peter was come to himself, he said, Now I know of a surety, that the Lord hath sent his angel, and hath delivered me out of the hand of Herod, and from all the expectation of the people of the Jews. -- Acts 12:7-11 (KJV)

Appeared to Herod

And immediately the angel of the Lord smote him, because he gave not God the glory: and he was eaten of worms, and gave up the ghost. -- Acts 12:23 (KJV)

Why people belief he is not Jesus

Appeared distinct from Jesus

"The Angel of the Lord" appeared to Joseph while Mary was pregnant with Jesus (Matthew 1:20-24).

And she shall bring forth a son, and thou shalt call his name JESUS: for he shall save his people from their sins. -- Matthew 1:21 (KJV)

Some would argue that Joseph was dreaming this and that is how it was possible for the Angel to be incarnated as a human fetus inside Mary while speaking to Joseph. However, the "Angel of the Lord" also appeared to the shepherds after Jesus was born as a distinct being and also refers to Jesus as another person:

And, lo, the angel of the Lord came upon them, and the glory of the Lord shone round about them: and they were sore afraid. And the angel said unto them, Fear not: for, behold, I bring you good tidings of great joy, which shall be to all people. For unto you is born this day in the city of David a Saviour, which is Christ the Lord. And this shall be a sign unto you; Ye shall find the babe wrapped in swaddling clothes, lying in a manger.

And suddenly there was with the angel a multitude of the heavenly host praising God, and saying, 14 Glory to God in the highest, and on earth peace, good will toward men. The shepherds glorify God.

And it came to pass, as the angels were gone away from them into heaven, the shepherds said one to another, Let us now go even unto Bethlehem, and see this thing which is come to pass, which the Lord hath made known unto us.

-- Luke 2:9-15 (KJV)

Unlike Joseph, the shepherds did not dream about the angel because they all witnessed it and spoke to each other.

Appeared not as Jesus said he would return

The Angel of the Lord also appeared the apostles at multiple accusations and to Herod after Jesus' ascension. This is not how Jesus said he would return.

Not identified as Jesus

Peter who was very familiar to Jesus and who was the first disciple to correctly identified him (Matthew 16:13-19) was also very familiar to the Angel of the Lord, such that Peter thought he was only seeing another vision. Yet, Peter identified "the Angel of the Lord" as a specific angel of God and not as Jesus (Acts 12:11). Surely Peter would not have mistaken the Angel of the Lord as someone else if he was Jesus.