In today's Gospel reading, Jesus warns us not to judge. He's speaking of the sin of pretending to be God who alone knows all that is in the hearts and motives and minds of sinners.
Judges sentence evil-doers, and God, who is the only completely holy judge, refrains from sentencing until the last possible moment. But you and I are quick to sentence people to all kinds of punishments (for example: "She doesn't deserve my time" or "He's entirely to blame and should fix this problem, not me").
However, we are supposed to judge between good and evil; this is the gift of discernment that comes from the Holy Spirit. And we are supposed to judge that a sinful act is evil and that a good deed is holy, as scripture elsewhere tells us to do. What we can never judge, because we are not all-knowing, is how fully sinners understand that what they're doing is wrong.
This warning from Jesus parallels the line from the "Our Father" prayer in which we ask God to "forgive us as we forgive those who sin against us". We can recognize when others have sinned against us and therefore need our forgiveness, but we must also recognize that we are sinners too and need to be forgiven.
If we judge others to be unworthy of our forgiveness, we're now committing a greater sin than theirs: We're condemning them to the absence of our love while glorifying ourselves as superior to them. Even worse, our pride is blocking God's love from reaching them through us.
Think of someone who often irritates you. Oh how we wish we could change people like that! We want them to realize that what they're doing is wrong. The truth is, we have ulterior, selfish motives: Of course we want them to change for their own benefit, but the reason we're irritated is because we want them to change for our benefit, to make our lives easier -- as if they're supposed to be the true source of our joy instead of God.
How can we help these people without sinning against them? Jesus explains it at the end of this Gospel passage. He doesn't prohibit us from recognizing the faults of others. He doesn't tell us not to remove the splinters in their eyes. What he does say is that we must first recognize our own sinfulness. Then we can approach them with humility instead of superiority.
Think of the various ways that others have aided you on your spiritual journey. Now imagine applying the same attitudes, the same types of words, the same patience, etc., toward anyone who might irritate you today.
Doing this mental exercise makes it easier to put Christ's command into action. It's a rehearsal that can help you handle situations with God's mercy instead of irritability, and with his ideas on how to help them.