Minecraft wiki entry for Player_versus_Player, associated text: Tournaments: Get a few players together: about 10+ is good, then put them two at a time against each other. If you win, you fight another winner; if you win again, you fight another winner again and again until you lose or there is only one winner left. You never fight a loser; if they lose they are out, except perhaps for a battle for third place.
Free-For-Alls/Last Man Standing: 3+ players fight against whoever they want in the arena at the same time. Maybe there is someone who is annoying or swears too much — who wouldn't want to try killing them…?
Mini team: Put around 5 players on equal teams (there can be more than two teams) and the last team standing wins.
Team: 10+ players on equal teams.
Un-official Duels: Let's say you have a person who you really hate, but you have more TNT than them? Call them on and if they say yes, start the battle. There is going to be a lot of fouls (like lava pouring, TNT and flint and steel burnings) but this is why it is an Unofficial duel, as they don't yet count.
Survival: Make about 10+ players and make monster spawners in the arena. Give all the players wooden swords. There should be no creepers. Last person standing wins. Players can attack their rivals to kill them or team up and share the victory.