Only 2 players face off at a time, it's a 1on1 duel to see not only who has the best aim, but also who can control the level the best. A big part of Tournament play is based around controlling the resources. If you can hog all the armor on the map, you should have an easy time killing your opponent.
Tournament mode is basically identical to a game of deathmatch with just two players.
This is a true test of skill because there are no outside forces to interfere with either player's game. Most Tournament games have a time limit instead of a fraglimit, because these games tend to be slower than the frantic pace of deathmatch. Long games commonly end with low scores because the players would rather retreat than risk dying. Tournament play is very difficult to master. The most successful Tournament players memorize the timing of every powerup on the level, so they can grab it the moment they pop up.