According to Dictionary.com

  1. characterized by uprightness or morality: a righteous observance of the law.
  2. morally right or justifiable: righteous indignation.
  3. acting in an upright, moral way; virtuous: a righteous and godly person.

William Tyndale (Bible translator into English in 1526) remodelled the word after an earlier word rihtwis, which would have yielded modern English rightwise or rightways. He used it to translate the Hebrew root צדק (tzadik), which appears over five hundred times in the Hebrew Bible, and the Greek word δίκαιος (dikaios), which appears more than two hundred times in the New Testament. -- Wikipedia

Righteousness is right standing, or good standing with God.

It does not mean human "perfection." You are righteous because God says you are.

You may wonder about Romans 3:10 which says no one is righteous. Yes, it does say that, but it is quoting an Old Testament passage to prove that both Jews and Gentiles were guilty before God and needed salvation. It immediately goes on to say in the next few verses (see Romans 3:22,24) that now we can be righteous through faith in Jesus Christ.

-- A Bible Devotion

Paul's explanation to the Romans:

Yet God, with undeserved kindness, declares that we are righteous. He did this through Christ Jesus when he freed us from the penalty for our sins. -- Romans 3:24 (NLT)

For the sin of this one man, Adam, caused death to rule over many. But even greater is God's wonderful grace and his gift of righteousness, for all who receive it will live in triumph over sin and death through this one man, Jesus Christ. -- Romans 5:17 (NLT)

For it is by believing in your heart that you are made right with God, and it is by confessing with your mouth that you are saved. -- Romans 10:10 (NLT)

Paul's explanation to the Corinthians:

God has united you with Christ Jesus. For our benefit God made him to be wisdom itself. Christ made us right with God; he made us pure and holy, and he freed us from sin. -- 1 Corinthians 1:30 (NLT)

For God was in Christ, reconciling the world to himself, no longer counting people's sins against them. For God made Christ, who never sinned, to be the offering for our sin, so that we could be made right with God through Christ. -- 2 Corinthians 5:19,21 (NLT)

Paul's explanation to the Galatians:

Yet we know that a person is made right with God by faith in Jesus Christ, not by obeying the law. And we have believed in Christ Jesus, so that we might be made right with God because of our faith in Christ, not because we have obeyed the law. For no one will ever be made right with God by obeying the law." -- Galatians 2:16 (NLT)

Paul's explanation to Titus:

... he saved us, not because of the righteous things we had done, but because of his mercy. He washed away our sins, giving us a new birth and new life through the Holy Spirit. -- Titus 3:5 (NLT)

Misconceptions about righteousness

Some people belief that you need to keep every law to be righteous, however Paul address this issue:

If righteousness comes through the law, then Christ died in vain. -- Galatians 2:21 (NKJV)

No one will be declared righteous in his sight by observing the law; rather, through the law we become conscious of sin. We all fall short of the glory of God. -- Romans 3:23 (NIV)