Multiplayer Modes
(partly copied from the internet and expanded on that basis)
Generally, you can divide multiplayer formats into three groups:
Everyone vs. Everyone
Everyone vs. Some
Team vs. Team
A rule that applies to all variants: When a player loses, all their permanents, hand cards, cards in the library, cards in exile, cards in the graveyard, and spells on the stack are removed from the game, and they can no longer play.
Everyone vs. Everyone
In "Everyone vs. Everyone" variants, all other players are opponents. A player only wins after defeating all other players.
This variant distinguishes between two forms - the normal game and Limited World (limited attack). In the normal game, you can attack any player, while in limited world, you can only attack your direct neighbors. With a high number of players, this can make the game much more manageable because you don't have to keep track of every single player. However, you can always only attack one player per turn. As the "target" of a spell, any player can be chosen.
With only three players, there is also the rule that you must attack your opponents in turn. This ensures that the game is fair and gains tactical depth.
Everyone vs. Some
The interesting aspect of "Everyone vs. Some" is that you can win by defeating only part of the other players. There are some players who are neutral and can only indirectly affect victory or defeat.
The most common form of this variant is Pentagram. In Pentagram (as the name suggests), five players play against each other. You win as soon as the two opponents directly opposite each other (on a pentagon) are defeated. This means that you never have your direct neighbors as opponents. But it can still sometimes be useful to play against your neighbors, especially if they are about to win. Pentagram is often played as 5-Color-Magic, where each player uses one of the five Magic colors (arranged like on the back of the cards); there is also the board game box Magic Game Night that I own. If the decks are well-balanced, very interesting games can emerge. Here too, you can only attack one player per turn and target any player with "target." Spells that mention "target opponent" or "opponents" only apply to the two opponents, not to the neutral players.
Team vs. Team
Team vs. Team is probably the most common multiplayer variant in Magic and the only one with officially established rules. The basic idea is that two (usually equally sized) teams are formed that then play against each other. Teams always win or lose together and are allowed to frequently consult with each other.
The classic Team vs. Team variant is such that two teams play against each other, and each has their own decks, life points, turns, etc. However, the game continues until all players of one team have lost, and the other team is the winner. To make the game more balanced and prevent a player from being eliminated too early, you can also play with shared life points (a team has a certain number of life points together) and/or the rule that all players of a team can block an attack.
Two-Headed Giant
Two teams of two players each play against each other. The respective players of a team are not independent but share life points and turns. Each team starts with 30 life points and loses as soon as the life points reach zero or one of the players loses the game for other reasons (e.g., because the library is empty). During a turn, both players can play and must consult with each other on what to do.
Emperor
Another popular team variant is Emperor. In Emperor, two teams with an odd number of players (usually three or five) play against each other. The middle player of a team is always the Emperor, the Imperator. As in chess, the game is lost as soon as the Emperor of the team is defeated. Since you can only attack your direct neighbor (which for Emperor is either one or none), the Emperor can only be attacked once all players on their side have been defeated. The remaining teammates are essentially his bodyguards, keeping opponents away from their Emperor for as long as possible. Additionally, the Emperor has 30 life points instead of the usual 20.