Judging is one of the most misunderstood concepts in the Bible. Many simply quote Matt 7:1, "Do not judge so that you will not be judged" and assume that is everything the Bible has to say about the topic of making moral evaluations. But they forget that this saying of Jesus was only the beginning of a paragraph (Matt 7:1-6) about making judgments and correcting others. Jesus taught that it is hypocritical to correct a friend's minor flaw while tolerating our own major flaws: "First take the log out of your own eye, then you will be able to see clearly to take the speck out of your brother's eye." His point is that it would be hypocritical (for example) to correct my friend's failure to tithe while tolerating my own financial dishonesty and lack of generosity towards those in need. And we should not forget that Jesus often judged - he regularly corrected both his opponents and his disciples.
Paul gives balanced advice for making judgments and correcting others in Galatians 6:1-5. "If someone is caught in some sin, you who are Spiritual should correct such a person in a spirit of humility..." Correction requires first making a moral evaluation of behavior based on the teachings of Scripture. The point of judging should only be to help others, not to make ourselves look good. Judging is wrong (and annoys us the most) when it is done in a prideful manner, or is done in a way to hurt people rather than help them.
Scripture regularly advises us to evaluate the character and actions of others. Take a look at 1 Thess 5:14, 2 Thess 3:14-15, James 5:19-20, Matt 18:15-20, Luke 17:3-4, 1 Cor 5:9-13, 1 Cor 6:1-6. In each of these, Christians are required to judge other Christians for the purpose of restoration. We cannot become more like Christ unless we are willing to humbly judge each other's behavior and "spur one another on toward love and good deeds" (Heb 10:24).
The picture: Christ the Judge, Michelangelo Buonarotti, 1537-41 (Sistine Chapel).