Team Deathmatch is the same thing except that now there are a few people whom you do not shoot. Capture the Flag is a Team Deathmatch representation of the classic kids game in which you have to infiltrate the enemy base, grab the flag, and bring it back to your base to score points. Tournament has two people fighting while other players watch, the victor fights the next opponent in line. In that bunch there is not one original concept, which makes Quake 3 essentially a game you've already played. Beyond that, there are virtually no gameplay tweaks available to add variety to the modes.
Every match is very similar.
Eventually it all becomes very mundane. The weapons don't really help. Again, you've seen each one before, and although the balance is very adept so that no one of them will rule the Arena, none of them are terribly impressive or subtle. Quake 3 is not a game of intellect, merely of twitch skills. The relatively small number of levels, along with the poorly programmed bots, make for a poor single-player experience.