The classic game mode requires that you eleminate all the other players meaning last man standing is the winner. There can be only one winner.
Team mode allows players to chose the team that they want to play on and the goal is to eliminate the opponent team players. The opponents are identified with a red marker above their head.
This means that you are willing to join any game type that is currently being played. This does not mean that the game type will randomly change during game play.