I'll get right to the point. Minigames are everywhere in every game, and Minecraft isn't an exception.
Minigame 1) Spleefing!
Simply put, you run around and break blocks beneath the feet of other people so they can fall into a pit of lava and die. This can be played in Classic and Beta. You win by being the last player standing.
Minigame 2) PVP
It's like in every other game. You grab a weapon and go to town. Kickin' players and takin' names is simpler than buttering bread. You lose by dying, and win by being the last player standing.
Minigame 3) Competitive Building
Minecraft is a phenomenon that is basically a game with lots of blocks that can be customized to allow you to build. So competitive building is building the best thing in a certain time with a few certain materials while following certain rules. Simple enough. You win by having the best stuff.
Minigame 4) Racing
Minecraft has sprinting, jumping, crouching, backpedaling, sailing, and carting. Design a combined racetrack and see who gets to the finish line first (that person is the winner).
Minigame 5) Hunting
Minecraft has mobs, both friendly and hostile. Grab a weapon and go to town on those skeletons. Whoever gets the most kills and/or the most loot wins. Pull a Dick Cheney and shoot another hunter (yes, I know that was an accident) if they're getting ahead if you want.
That's all for now, folks! Add your own minigames in the comments, I know there are like 20 more.