There are several distinct types of Multiplayer modes in Video Games. This trope is a Sister Trope of Meta Multiplayer, Co-Op Multiplayer and Massively Multiplayer. See also Misbegotten Multiplayer Mode, where multiplayer modes are featured in games that are really designed for single-player.
CompetitivePlayers directly compete against each other (Player Versus Player) and possibly characters controlled by the CPU. May organize players into teams. Usually the main feature of Fighting Games, sports games, Racing Games, party games, puzzle games, and anything online, though a single-player (or co-op) "main game" may be available as well.
2 PlayerWhen the game is only designed for one player against another. Modes may be provided where additional players can play alongside a comrade, but if one person plays against another it would not be considered two teams of one. Examples:
Counter-OpThe second player takes control of enemies and has no objectives other than to hurt the first player, who plays exactly as in single-player mode.
Red Versus Blue2 equivalent teams with the same capabilities and objective play against one another. These teams may be comprised of only one player, but teams scale naturally. Teams are often (but not necessarily) distinguished by looking identical except for being colored either Red and Blue (although note different, asymmetric types of 2-team multiplayer may use these to distinguish teams as well). Examples:
Attack and DefenseTwo teams have different objectives. Examples:
Commander and FightersOne player on a team interacts in the game in a diferent fashion from the rest, eg. a Commander with a Real-Time Strategy view of the battlefield while the rest of the team fights in a First-Person Shooter view. Examples:
3 or 4 sided
4-SquareThe playfield is set up for no more than 4 players, eg. each player gets a corner of the playfield. Most commonly seen in the days of the N64 when systems could have no more than 4 players at any one time. If there are three players/teams, one team is visibly absent. Examples:
1 vs. 3One player/team must fight another team of 2 or 3 other teams/players. The game is balanced so the single player is in a position of power. Examples:
Free-For-AllThe game has as many competing groups as the player count permits.
Player Versus EveryoneEvery player plays against every other player. Examples:
Multi-TeamPlayers can cooperate in their own sets of groups in any permutation allowed by the player limit. Examples: