judge: to form an opinion about through careful weighing of evidence and testing of premises.
Simply stated, judging is the process of evaluation that people naturally employ everyday to make determinations of what is true.
We judge when we need to leave to arrive on time to a particular destination. We judge what we should wear based on what the weather is for the day. We judge the credibility of others as they give us information, etc.
In Scripture, there are many references to the process of judging that may not specifically use the word judge. For example, the words translated as "judge" in Scripture are also translated examine, search, discern, ask, question, contend, esteem, and determine.
The problem arises because judging can be subjective. In other words, your judgments are based on your perceptions, motives, and understandings. This is why two people can judge the exact same situation and come up with different determinations.
We know from Scripture that God is the Judge of all (Genesis 18:25; Judges 11:27; I Samuel 2:10 Psalms 50:6; Psalms 96:13; Psalms 98:9; Isaiah 3:13; Isaiah 33:22; Jeremiah 11:20; Ezekiel 18:30; Ezekiel 33:20; Hebrews 12:23; I Peter 1:17; Matthew 12:27).
God is set in position as our Judge because He is all-knowing, He is Truth. As such God's judgments are righteous and true (John 8:26; Romans 11:33; Revelation 16:7; Revelation 19:2).
The question then becomes, who is equipped to proclaim God's judgments?