Deathmatch
Deathmatch is a widely used gameplay mode very well integrated into First-person shooter computer games and originally coined by John Romero in 1993 during the development of DOOM. The target of a deatchmatch game is to kill (or "frag") as many other players as possible in the time given; until the level is completed or until the frag limit or time limit is reached. Once one of these conditions is met the match is over and the winner is whoever has accumulated the most frags.
Team Deathmatch is like the above, but with players organised into 2 or more teams, with each team having its own frag-count. Friendly fire may or may not cause damage, depending on the game and the rules used - if it does, a player killing a team-mate usually decreases the team's score by 1.