A judge is a figure of authority, one who interprets the laws of a culture and often sentences those who have been found guilty of breaking the laws.
The exact role of the judge varies across cultures. In Cardassian society, the role of judge is filled by the Chief Archon. (DS9: "Tribunal") In most Federation societies judges have filled important roles throughout history and many judges, such as T'Pau and Aaron Satie, are revered figures in their cultures. (VOY: "Darkling"; TNG: "The Drumhead")
Judges may also act as arbiters in contests, using their own judgement to subjective qualities to determine the winner. Emony Dax was a judge in several gymnastics competitions. (DS9: "Trials and Tribble-ations")