What does ‘do not judge’ mean?

“Assuredly, I say to you, among those born of women there has not risen one greater than John the Baptist; but he who is least in the kingdom of heaven is greater than he.”

(Mt 11:11)

You who love the Lord, hate evil!
He preserves the souls of His saints;
He delivers them out of the hand of the wicked.
(Psalm 97:10)

Hate evil, love good;
Establish justice in the gate. (Amos 5:15)

A righteous man hates lying,
But a wicked man is loathsome and comes to shame.
(Proverbs 13:5)

[There is] A time to love, And a time to hate.
(Ecclesiastes 3:8)

But Jesus went to the Mount of Olives. Now early in the morning He came again into the temple, and all the people came to Him; and He sat down and taught them. Then the scribes and Pharisees brought to Him a woman caught in adultery. And when they had set her in the midst, they said to Him, “Teacher, this woman was caught in adultery, in the very act. Now Moses, in the law, commanded us that such should be stoned. But what do You say?” This they said, testing Him, that they might have something of which to accuse Him. But Jesus stooped down and wrote on the ground with His finger, as though He did not hear. So when they continued asking Him, He raised Himself up and said to them, “He who is without sin among you, let him throw a stone at her first.” And again He stooped down and wrote on the ground. Then those who heard it, being convicted by their conscience, went out one by one, beginning with the oldest even to the last. And Jesus was left alone, and the woman standing in the midst. When Jesus had raised Himself up and saw no one but the woman, He said to her, “Woman, where are those accusers of yours? Has no one condemned you?” She said, “No one, Lord.” And Jesus said to her, “Neither do I condemn you; go and sin no more.”

(John 8:1-11)

Go and sin no more.

(John 8:11)

In Jesus Christ, it is God himself who goes in search of the “stray sheep”, a suffering and lost humanity.

(BENEDICT XVI, Encyclical Letter Deus Caritas Est, 12)

“Hate the sin and love the sinner.”

For a good many people imagine that forgiving your enemies means making out that they are really not such bad fellows afer all, when it is quite plain that they are. Go a step further. In my most clear-sighted moments not only do I not think myself a nice man, but I know that I am a very nasty one. I can look at some of the things I have done with horror and loathing. So apparently I am allowed to loathe and hate some of the things my enemies do. Now that I come to think of it, I remember Christian teachers telling me long ago that I must hate a bad man’s actions, but not hate the bad man: or as they would say, hate the sin but not the sinner.
For a long time I used to this a silly, straw spilling distinction: how could you hate what a man did and not hate the man?
But years later it occurred to me that there was one man to whom I had been doing this all my life – namely myself.
However much I might dislike my own cowardice or conceit or greed, I went on loving myself. There had never been the slightest difficulty about it. In fact the very reason why I hated the things was that I loved the man.
Just because I loved myself, I was sorry to find that I was the sort of man who did those things.
Consequently, Christianity does not want us to reduce by one atom the hatred we feel for cruelty and treachery. We ought to hate them. Not one word of what we have said about them needs to be unsaid. But it does want us to hate them in the same way in which we hate things in ourselves: being sorry that the man should have done such things, and hoping, if it is anyway possibile, that somehow, sometime, somewhere he can be cured and made human again.

(CLIVE STAPLES LEWIS, Mere Christianity, Harper Collins, 2012, p. 117)

It cannot escape our minds when facing a topic like this, the words of Jesus, ‘Do not judge, and you will not be judged’ (Luke 6:37): evaluation would seem the opposite, evaluation is judging. In what sense did Jesus invite not to judge?
[…]
Let’s try to understand better.
Judging is the most human act there is: animals do not judge, but humans do. So, we are called to judge, and evaluation is the name given in schools to an act of judgment: the assertion of a value through tools. I was wondering why Jesus says in the Gospel that we should not judge—especially since, in other instances, he says that we should judge, and St. Paul says, “Test all things; hold fast what is good” (1 Thessalonians 5:21). Do not judge so that you will not be judged. It’s like saying: be careful because God will do the math.
Your task—in your relationships with each other—is to judge, but not in the sense of condemning. Be careful that judgment is never a closure, a drawing of a line, a departure from hope. Never allow yourself to say, “It’s over, there’s nothing to be done with this.” But we shouldn’t be confused: it doesn’t mean that, in the end, one doesn’t fail or give a grade of four; it means that even if we fail or give a four, we do it truly to accompany the other. Even the four can contain an opening, and in fact, if I give a four, it pushes me to do everything to understand how the other can take a step.

(FRANCO NEMBRINI, Di padre in figlio. Conversazioni sul rischio di educare, Ares, Milano, 2011, p.180-181)

Respect and love ought to be extended also to those who think or act differently than we do in social, political and even religious matters. In fact, the more deeply we come to understand their ways of thinking through such courtesy and love, the more easily will we be able to enter into dialogue with them.
This love and good will, to be sure, must in no way render us indifferent to truth and goodness. Indeed love itself impels the disciples of Christ to speak the saving truth to all men. But it is necessary to distinguish between error, which always merits repudiation, and the person in error, who never loses the dignity of being a person even when he is flawed by false or inadequate religious notions.

(Constitution Gaudium et Spes, 28)

The Church is uncompromising on principles because it believes; it is tolerant in practice because it loves. The enemies of the Church, on the other hand, are tolerant on principles because they do not believe, but uncompromising in practice because they do not love. The Church absolves sinners; the enemies of the Church absolve sins.

(RÉGINALD GARRIGOU-LAGRANGE, Dieu, son existence et sa nature, Paris, 1923, p. 725)