Each Judge runs their own type of games; it's always some type of competition, but it can range from silly to instructional to lethal, depending on who's running it. For the latter, several of them enjoy pitting teams against each other in various ways. The important thing to remember is that what happens in the Arena, stays there, if we can manage it. Some teams can't manage to follow that rule, but Canis does.
There are twenty-eight Judges that we know of, plus three destroyed or inactive. In no particular order: King, Queen, Messenger, Beauty, Persona, Storyteller, Mirror, Game Master, Beast, Flowers, Haunted, Veiled, Angel, Clockmaker, Foreman, Gambler, Lover, Shadow, Soldier, Dude, Kid, Twins, Graces, Waterbearer, Huntress, Horn, Guardian and Librarian. Of those, the ones known to regularly be dangerous are Huntress, Game Master, Flowers, Storyteller, Persona, Mirror, and Horn in combination with certain others. The rest may or may not be, but tend towards harmless. Guardian watches over the Judges' enforcers, Ophiuchus; he doesn't run games, and neither does Beast. Waterbearer may or may not still be alive and sane, at this point; we don't know.