A poker tournament, like any other tournament, ends with just one man standing, and as with other tournaments your prize depends on how well you place and the prize pool is (usually*) made up of the money that you pay to enter the tournament. For instance, if the buy-in is $50, and 20 people enter, the total prize pool would be $1,000. The pay-out structure of the tournament (how many get paid, and how much each place pays) varies both with number of entrants and from site to site.
Finishing in the money, meaning that you place high enough to get rewarded, typically wins you at least a little bit more back than you paid to enter, but not by much. The big pay-outs usually begin with the final table, or (in the case of smaller tournaments) with the top three players.
When you enter an online (well, any) poker tournament, you have to finish it - you will not get reimbursed in case you have to leave halfway. You play until you're knocked out or until you've won. When the tournament starts, everyone gets a stack of tournament chips to play for (these stacks contain the same amount of chips for everyone of course), and because they have no real value outside of the tournament, they can have a large dollar value even if the buy-in is small. For instance, a $5 tournament on PokerStars may have a $1,500 starting stack, just as a $500 tournament will. You will sometimes see tournament stacks referred to as "T5000" rather than "$5000"
Unlike cash poker games, there is no rake on the pots that are played. Instead, the sites make their money off of tournaments by adding a bit extra to the buy-in. This extra fee is usually around 10% of the buy-in, but can vary. When speaking of tournament buy-ins, therefore, people often specify a tournament entry cost as this: $[buy-in]
Because online tournaments take no effort to organize, they are always available, around the clock. All it takes is waiting for, say, 10, or 30, or 45 or 180 (or however many the tournament is set to contain) people to pay the entry fee and it can start (these tournaments are called "sit&go"-tournaments, or SnGs) - no need to schedule a time and a place. This, in turn, has made tournaments a viable way of making money off of poker where before they were rather considered icing on the cake, and there are online poker professionals who exclusively live off of tournaments. There are both benefits and drawbacks to choosing this particular venue of poker; the upside is that tournaments - generally speaking - are more complex than a standard cash game. This in turn means that skilled players will have a bigger edge over the poor players, meaning that they have an easier time making money. The drawback, however, is that you only need to be unlucky once in a tournament to not make any money whatsoever. For players who spend most of their time playing large multitable tournaments, there could be (relatively) long times between finishing in the money. Of course, winning a large tournament usually means a huge pay-out.
A special type of tournament, called a "satellite," merits its own paragraph: A satellite is a tournament where only one person wins a prize - or all the people who win, win the same prize - and this necessitates a different strategy compared to a regular tournament. The most common kind of satellite is the one where first place (or a specific number of top places) is rewarded with a buy-in for a bigger tournament. It was by winning satellites, for example, that Chris Moneymaker managed to go all the way to win the WSOP Main Event, not having to pay the $10,000 entry fee himself.
When it comes to rules of play in a tournament compared to cash games, the most notable difference is that in a tournament, the size of the blinds will increase. Poker sites handle this differently, but PokerStars, for instance, increase their blinds after a certain amount of time. Other sites increase the blinds after a certain number of hands. Making the blinds go up is a necessity, because the winning players will accumulate more and more chips, and having a stack of 1.2 million, with blinds at 10/20, is a bit silly.It's also a way to force action; to punish the passive. In a tournament, unless it's played with very deep stacks, you cannot sit back and wait only for premium hands - you will have to gamble with more speculative holdings from time to time. Learning when and how to do this is a major part of tournament strategy. But I digress; this is not a strategy guide.
Although no-limit tournaments are generally the most popular (and specifically no-limit hold'em), you will also find limit and pot-limit tournaments offered, for hold'em as well as omaha or stud, or other variants. Most sites have trouble filling up the SnG tournaments for anything but no-limit hold'em, however, so if you're looking to play a pot-limit Omaha8 SnG, you may have to be prepared to wait awhile before it starts.
Poker Tournaments, unless otherwise stated, are so called freeze-outs, meaning that if you lose all your chips, you're out. There are, however, also tournaments that allow for rebuys, meaning that if you get knocked out, you can pay a fee to get back in the game, with a new starting stack. The rebuys are only allowed for the first part of the tournament (stopping after a certain amount of hands, or a certain amount of time, typically), and in some tournaments you are only allowed to rebuy a certain number of times. Different sites and tournaments have different rules; make sure you know which ones apply to you before you sign up.
As rebuys have been mentioned, so should add-ons. Some tournaments allow for add-ons, which are similar to rebuys, but you don't need to get busted to get more chips. You can simply pay a fee to get extra chips added to your stack. As with rebuys, the add-ons are almost always restricted by time after tournament start and/or number of times that you can purchase them.* There are exceptions, most notably "freerolls" - where it costs nothing to enter, but the site (or Cardschat.com, for example) sponsors the prize pool.