There are two types of tournaments, Limited and Constructed. Limited means you play with a limited card pool, cards that are provided by some means on the premises where the tournament is being held. Constructed means that you play with decks made out of cards that you own, and constructed beforehand, that you bring to the tournament with you.
Within Limited, there is Draft and Sealed. Both are played using unopened booster packs. Sealed is where each player is given 6 or so booster packs, and the cards from all of those form his sealed pool, which he then uses to construct a 40-card minimum deck with which to play. Players are given a time limit for opening their cards and constructing their decks before the tournament begins.
Draft is a type of format with several different variations, but all consist of players each using a few booster packs (usually 3 each), with each player opening one of their packs, selecting a card from it, and then passing that pack to another player. In this way, each player drafts (or picks) his deck one card at a time from the available packs.
Within Limited, there event will specify which set is being used, and will use booster packs from that set. A sealed pool will usually be from all of the same set, whereas a draft will usually be one pack from every set of a block (a block consists of 3 sets) that have so far been released, e.g. Scars of Mirrodin/Mirrodin Besieged/New Phyrexia, or M12/M12/M12.
Within Constructed, the type of format of the tournament dictates which cards can be used.
Standard allows the use of the most recent core set, and the most recent two blocks (3 sets each), with a narrow window of eligibility of the previous core set between the release of a new core set and the next new block.
Vintage and Legacy both allow the use of cards that have been printed earlier than those legal for use in standard. You can read more about the various formats here.