With all of the forms of poker being played, tournament poker is by far the most popular. All of the televised poker action, from the World Series of Poker to the National Heads Up Championship, is tournament poker. And you can find tournament action online, any time.
If you know what kind of tournament you're interested in playing, you can use the tournament Finder on the right, under the heading "Poker Tournaments". The tournament finder will be displayed in a pop-up window, so a pop-up blocker may interfere with starting it. If this happens, hold down the Ctrl key when clicking the link (located bottom left of your keyboard).
You can use the tournament finder to search for specific games, limits, and stakes. You can look for freerolls, Omaha Hi-Lo pot limit, 7 Stud fixed limit, whatever you're interested in playing. The finder will tell you when the tournament starts, and the cost to buy in. It also gives a description of the game, letting you know whether it's a guaranteed money game, a freeroll, or a standard tournament.
On the right there are links to pages covering some of the larger events in the poker world, including the World Series of Poker, the events on the World Poker Tour, and the College Poker Championships.
In case you're not familiar with the term freeroll, a freeroll is a poker tournament where there is no entry fee, but the prizes for the winners are real. I have a page dedicated to freerolls with more information on them.
Also online you'll find something called the Sit and Go tournament. These are usually single table tournaments (but some rooms do have multi-table SnG's). Instead of starting at a set time, they run all day. Each tournament begins as soon as all the chairs are filled. That makes playing tournament poker online about as convenient as possible. Find a table, sit down, and go.