Nonjudgment is the intentional ceding to God the sole role of judge; we refuse to take God's judgment into human hands. Nonjudgment lives in submission to Jesus' explicit teaching in the Sermon on the Mount.
Nonjudgment is a "paying forward" of the grace a mercy received by the sinner from Christ; by practicing nonjudgmental we preach the gospel (John 20.23). Improvising on this text, the Desert Fathers say: "When we cover a brother's sin, God covers our sin; when we tell people about our brother's guilt, God does the same with ours."
This is the strange logic of the gospel--the opposite way.
We refuse to judge just as we refuse to engage in violence--so that we are vanguards, heralds, the New Adam/Eve of a wholly new way on earth and refuse to perpetuate what tearing us apart.