What does Jesus mean "Do Not Judge"?

"Do not judge" (in Matthew 7:1-5) is probably one of the most quoted but misquoted words of Jesus. It is often used to tell someone not to tell someone else what is wrong with his/her behaviour. I have probably said it myself. Or I have felt guilty at times for having thought or said something negative towards others. More recently, I have also adopted a new attitude that I don't pass judgement outside the church but it is my responsibility to admonish someone who is my brother or sister in Christ. This is based on St. Paul's teaching in 1 Corinthians 5:12-13a "For what have I to do with judging those outside? Is it not those who are inside that you are to judge? God will judge those outside." So I started digging deeper into this and here is my new discovery:

(1) The Ignatius Catholic Study Bible has surprisingly little to say on Matthew 7:1-5. It says Jesus condemns judging other's faults because we are incapable of judging with fairness and accuracy. However, Jesus commands us to exercise critical discernment (Matthew 7:6, 15-19, 1 Thess 5:21). We should set the standards of our personal judgement by our own conduct towards others.

(2) Gotquestions.org is quite helpful. It points out that "the Bible’s command that we not judge others does not mean all actions are equally moral or that truth is relative. The Bible clearly teaches that truth is objective, eternal, and inseparable from God’s character. Anything that contradicts the truth is a lie—but, of course, to call something a “lie” is to pass judgment. To call adultery or murder a sin is likewise to pass judgment—but it’s also to agree with God. When Jesus said not to judge others, He did not mean that no one can identify sin for what it is, based on God’s definition of sin.". It quotes Jesus' words "Do not judge by appearances, but judge with right judgment." (John 7:24) and list these sinful type of judgement:

• Superficial Judgement - Passing judgment on someone based solely on appearances before investigating the facts, e.g. Luke 7:36-50. 
• Hypocritical Judgement - When we point out the sin of others while we ourselves commit the same sin, we condemn ourselves (Romans 2:1).
• Harsh, unforgiving judgment - We are “always to be gentle toward everyone” (Titus 3:2). It is the merciful who will be shown mercy (Matthew 5:7), and, as Jesus warned, “In the same way you judge others, you will be judged, and with the measure you use, it will be measured to you” (Matthew 7:2).
• Self-righteous judgment - We are called to humility, and “God opposes the proud” (James 4:6). The Pharisee in Jesus’ parable of the Pharisee and the tax collector was confident in his own righteousness and from that proud position judged the publican; however, God sees the heart and refused to forgive the Pharisee’s sin (Luke 18:9–14).
• Untrue judgment - The Bible clearly forbids bearing false witness (Proverbs 19:5). “Slander no one” (Titus 3:2).

The article then gives a nice summary: "Believers are warned against judging others unfairly or unrighteously, but Jesus commends “right judgment” (John 7:24). We are to be discerning (Colossians 1:9; 1 Thessalonians 5:21). We are to preach the whole counsel of God, including the Bible’s teaching on sin (Acts 20:27; 2 Timothy 4:2). We are to gently confront erring brothers or sisters in Christ (Galatians 6:1). We are to practice church discipline (Matthew 18:15–17). We are to speak the truth in love (Ephesians 4:15)."

(3) Catholic Answers has a very nice article by Jim Blackburn with very similar teaching as (2) but includes lots of examples, like this one: "Typical examples would be dealing with an adult child who’s living with her boyfriend or coping with an adult sibling who has announced that he is gay. The callers often struggle with whether to allow the child or sibling to practice the immoral lifestyle in their home. Do I have to let them spend the night? What do I tell my kids? How do I deal with this in a loving way? Can I truly love my neighbor while rejecting his immoral lifestyle?" After giving a very detail explanation on Matthew 7:1-5 he gives this conclusion:
"So, when faced with the immoral behavior of loved ones, how can we be sure to rightly judge behavior? In Jesus’ own words, we must start by taking the logs out of our own eyes—by making sure we are doing the best we can to live lives of good example. We must also strive to form our consciences correctly so that we know sin when we see it. Finally, we must not jump to conclusions about another’s culpability in sin. Doing all this will help to ensure that our admonitions are seen as the loving actions we intend them to be—meant to help our loved ones live their lives in ways that are pleasing to God. Only then can our efforts be effective in helping to take these ugly specks out of our brothers’ eyes."

(4) Tim Staples puts on a positive spin on judging others: Jesus Commands Christians to Judge!
"For those who misunderstand Matt. 7:1, this sounds like a contradiction. But if we pause to think about this for a moment, the spiritual works of mercy we should have been taught from our childhood represent the classic example of this truth. And these are not optional, folks. Our eternity depends upon fulfilling them in our lives."

What are the Spiritual Works of Mercy (see CCC 2447)?
1. To comfort those who are suffering.
2. To console those who have suffered loss.
3. To forgive those who ask for pardon.
4. To forbear with those who hurt us, even if they are not sorry.
5. To admonish the sinner
6. To educate the ignorant.

The last two—to admonish the sinner, and to educate the ignorant—necessarily involve making judgments concerning the actions of others. See these biblical teachings:

"My brethren, if any one among you wanders from the truth and some one brings him back, let him know that whoever brings back a sinner from the error of his way will save his soul from death and will cover a multitude of sins." (James 5:19-20)

"If I say to the wicked, "You shall surely die," and you give him no warning, nor speak to warn the wicked from his wicked way... that wicked man shall die in his iniquity; but his blood I will require at your hand." (Ez. 3:18-19)

"Take heed to yourself and to your teaching; hold to that, for by so doing you will save both yourself and your hearers." (I Tim. 4:16)