“Judge not, that you be not judged,” does not mean, as the world likes to pretend, “Christians aren't ever allowed to judge anyone under any circumstances ever, but we get to judge them for judging.” Christ’s words here are not a get-out-of-judgment-free card when Christians look negatively upon your promiscuity or vulgarity or cruelty, as so many people with a very shallow understanding of the Christian faith like to claim. This is all quite obvious to anyone who’s read the rest of the Bible and noticed that there are about 4 billion instances when God commands His people to judge others.
Rather, what Jesus is telling us with these words is that the kind of judgment you heap upon others, God will heap upon you. If you remove the log from your own eye first, if you ensure that you can see clearly to lead people to repentance and trust in Christ, then you should expect the same from God. If your judgment leads people to Christ’s forgiveness, then Christ Himself will lead you to His forgiveness. But if you are only concerned with tearing your neighbor down and not at all concerned with your own sin, then you should expect the opposite. If you have no interest in removing the log from your eye and leading your neighbor to Christ, then God will see your judgment for what it is: trust in your own righteousness and a rejection of the righteousness of Christ. So in your judgment of your neighbor, if you treat him like Christ doesn’t exist, that’s how your Father in heaven will treat you. He will not give you the merits of Christ’s salvation. He will judge you according to your own works and give you the verdict you've earned. He will point to the log in your eye and condemn you for eternity.
So don’t judge those God hasn’t given you to judge. Leave them to the authorities God has placed in their lives. And when it comes to those who have been given to you, judge them with your eyes on Christ, leading them to the blood He poured out for them and for you. Likewise, when your neighbor stumbles into sin and needs a clear word, show him how you removed the log from your eye by looking to the wood of the cross. Show him how to wash away the speck that blinds him by looking to Christ who always opens the eyes of the blind.