What Does "Judge Not Lest Ye Be Judged" Really Mean?

Luke 6:37-38

³⁷ “Judge not, and you shall not be judged. Condemn not, and you shall not be condemned. Forgive, and you will be forgiven. ³⁸ Give, and it will be given to you: good measure, pressed down, shaken together, and running over will be put into your bosom. For with the same measure that you use, it will be measured back to you.” 

So is Jesus contradictory when he says the following?

Leviticus 19:15-16

¹⁵ ‘You shall do no injustice in judgment. You shall not be partial to the poor, nor honor the person of the mighty. In righteousness you shall judge your neighbor. ¹⁶ You shall not go about as a talebearer among your people; nor shall you take a stand against the life of your neighbor: I am the LORD. 

And

Leviticus 20:1-3

 ¹ Then the LORD spoke to Moses, saying, ² “Again, you shall say to the children of Israel: ‘Whoever of the children of Israel, or of the strangers who dwell in Israel, who gives any of his descendants to Molech, he shall surely be put to death. The people of the land shall stone him with stones. ³ I will set My face against that man, and will cut him off from his people, because he has given some of his descendants to Molech, to defile My sanctuary and profane My holy name.

Leviticus 19:15-16 calls for that "[y]ou shall do no injustice in judgment... In righteousness you shall judge your neighbor". Meanwhile, Leviticus 20:1-3 calls for that "'[w]hoever of the children of Israel, or of the strangers who dwell in Israel, who gives any of his descendants to Molech, he shall surely be put to death.'" Isn't that judging (Leviticus 19:15-16) and taking a stand against the life of a neighbor (Leviticus 20:1-3)?

So what does "Judge not" mean, besides that Jesus is not at all contradictory? Don't be a hypocrite! After all, "...He repays man according to his work, And makes man to find a reward according to his way." (Job 34:11); and "...if anyone builds on this foundation with gold, silver, precious stones, wood, hay, straw, each one’s work will become clear; for the Day will declare it, because it will be revealed by fire; and the fire will test each one’s work, of what sort it is." (1 Corinthians 3:12-13)

So, judge but don't be a hypocrite. "[']For with the same measure that you use, it will be measured back to you.'" Paul then was not contradictory when he said the following about a young Corinthian national:

1 Corinthians 5 (Emphasis mine)

Immorality Defiles the Church

 ¹ It is actually reported that there is sexual immorality among you, and such sexual immorality as is not even named^[a] among the Gentiles—that a man has his father’s wife! ²And you are puffed up, and have not rather mourned, that he who has done this deed might be taken away from among you. ³ For I indeed, as absent in body but present in spirit, have already judged (as though I were present) him who has so done this deed. ⁴ In the name of our Lord Jesus Christ, when you are gathered together, along with my spirit, with the power of our Lord Jesus Christ, ⁵ deliver such a one to Satan for the destruction of the flesh, that his spirit may be saved in the day of the Lord Jesus.^[b]
6 Your glorying is not good. Do you not know that a little leaven leavens the whole lump? ⁷Therefore purge out the old leaven, that you may be a new lump, since you truly are unleavened. For indeed Christ, our Passover, was sacrificed for us.^[c] 8 Therefore let us keep the feast, not with old leaven, nor with the leaven of malice and wickedness, but with the unleavened bread of sincerity and truth.

Immorality Must Be Judged

   
⁹ I wrote to you in my epistle not to keep company with sexually immoral people. ¹⁰ Yet Icertainly did not mean with the sexually immoral people of this world, or with the covetous, or extortioners, or idolaters, since then you would need to go out of the world. ¹¹ But now I have written to you not to keep company with anyone named a brother, who is sexually immoral, or covetous, or an idolater, or a reviler, or a drunkard, or an extortioner—not even to eat with such a person. 
¹² For what have I to do with judging those also who are outside? Do you not judge those who are inside? ¹³ But those who are outside God judges. Therefore “put away from yourselves the evil person.”^[d]