There are several types of tournament environments sanctioned by the
DCI in the game of Magic, and the environment you're playing in determines which cards are legal.
Constructed Formats
A constructed deck needs a minimum of sixty cards, with an optional fifteen-card sideboard. This is the commonly known version of Magic, where people build decks at home, then square off against each other. With the exception of basic lands, you may run no more than four of any given card.
Type II (also known as Standard): In this environment, you can use any card from the most recent base set (i.e. 8th, 9th), and the two most recent "blocks" of expansion sets. This is by far the most popular environment, mostly due to the fact that this format has the easiest cards to acquire and the large events that take place. This is the format that makes the money for Wizards of the Coast, so they run large events such as the Grand Prix and the Pro Tour, where professional players can make large sums of money.
Type I (also known as Classic or Vintage): In this environment, you can use any card ever printed in an expansion or base set, as well as many of the promotional cards. Notable exceptions are the ante cards and the entire Unglued expansion set, which was only a joke, anyways. Type I is notorious for first-turn kills, although they are a lot rarer than most people think. Those are just the very memorable games. This environment contains a lot of the broken cards such as the Black Lotus, an artifact that comes out for free and gives you three extra mana, allowing for explosive starts. There are only so many of these older cards to go around, though, so they tend to fetch rather large prices on E-Bay. Some of the more powerful cards are "restricted", which means that you may only run one of them in a deck.
Type 1.5 (also known as Classic Restricted): A good idea, but not played anywhere near as much as Type I. Type 1.5 is Type I, but all of the cards that are restricted in Type I are just outright banned, instead. It's a slower, lower-power format toted as the solution to the "overpowered" Type I environment that just never really caught on. The DCI recenty remodeled 1.5 into its own format, with its own banned list. All of the fast combos have been banned, leaving what's shaping up to be a very healthy-looking format. They also seem to be throwing more support behind it.
Type 1.X (also known as Extended): Extended lets you use any card from 6th Edition or newer, as far as base sets go, and from any expansion from Tempest on. Extended lets you use any card set from 7th Edition or newer, and from the expansion set Invasion or newer. The extended field is quite a bit larger than the Type I field, but nowhere near as large as the Type II field. Type II decks tend to become Type 1.X decks as their cards rotate out of the Type II format.
Block Constructed: Everyone decides on a block to use, then builds a deck using only cards from that block. A variation of this called BYOB lets you create your own block, using the first set from one block, the second set from another, and the third set from a final block.
Limited Formats
These formats allow you to use forty-card decks, but you build your deck at the event, using only cards you receive as part of the tournament.
Sealed: Everyone gets a seventy-five-card tournament pack from the first expansion of the block, consisting of thirty basic lands (six of each color) and forty-five other cards, as well as two fifteen-card booster packs. If there's only one expansion in the current block, both booster packs are that expansion. If there are two expansions, then both boosters are the second expansion. If there are three expansions, then the booster packs consist of the second and third expansion. Once you've opened your cards, you build a deck of at least forty cards. Everything you don't use is your sideboard.
Draft: Drafts usually take eight people. Unofficial drafts can be run with fewer. Everyone gets three booster packs. There are various kinds of drafts, but here's an example of one: Everyone takes their first pack and opens it. They look through it, decide which card they want, then pass the remaining cards to their left and repeat until no cards are left. Then, they open their second pack and repeat, but going to the right. Finally, the third pack is opened, and passed to the left. People can grab as many basic lands as they want, and then they build a forty-card deck that they use to compete, with the leftover cards forming the sideboard.
Terms:
Sideboard: An optional fifteen cards that are included with your deck. During the first game of a match, you must use your original deck, but before the second (or third or fourth) game of the match, you may switch cards in your sideboard for cards in your deck on a one-to-one basis. This allows you to include some cards that allow you to deal with troublesome decks without getting stuck with useless cards in other match-ups.
Block: Magic expansion sets are grouped into sets of three. The two most recent blocks are the Odyssey block, consisting of Odyssey, Torment, and Judgement, and the Onslaught block, consisting of Onslaught, Legions, and a set that has not been released yet.
Ante cards: Cards that involve "anteing" a card from your deck, as sort of a forced trading.
DCI: Duelist's Convocation International, the governing body of tournament play in the game of Magic: the Gathering.
Current banned/restricted lists are available at: http://www.crystalkeep.com/magic/rules/tournament.html