There are 13 events across the Debate, Congress, and Speech (rhetorical and interpretation) categories. See this list describing the events and who would like each of them. Most students try more than one event before finding their favorite!
The best competitors pick one event to specialize, but you don't have to specialize at the start. Try a few things, figure out what you like, and then do that one more often. Speech competitors can, at most big tournaments, enter multiple events at the same time!
You can see the full schedule here (with more links on the home page). Unlike a sport, you can compete whenever and as often as you want to, and it doesn't hurt anyone else. However, it is always more fun with more people at a tournament, and the more you compete the more you learn!
Our tournament schedule runs from September to June. Regular season tournaments, which are open to anyone, run from September to February. Starting in March, there are some more restrictions on some tournaments.
We travel as a team, but each entry at a tournament competes separately. That means you won't compete against your teammates (except at very small tournaments) and your performance doesn't affect anyone else.
At a tournament, everyone competes at the same time in different rooms across the site. That way, everyone who goes to the tournament will compete several times. What "compete" means depends on the event: in Speech, a room of 4-6 competitors will perform for a judge, who will then rank the competitors; in Debate, two teams/debaters will debate in front of a judge, who will pick a winner; in Congress, 12-20 competitors will give speeches in front of judges, who will then rank the competitors.
The rounds where everyone competes are called preliminary rounds. At our local tournaments, everyone will compete a few times, then we get some food, and then we head home with ballots - essentially the score sheets from the judges that give feedback on the performance/debate. At larger tournaments, the top competitors will advance (or "break) to elimination rounds, which will narrow down the field to a final round and declare a champion. Those are great opportunities to earn trophies or to get to watch and learn from the best.
And then the next weekend, we do it all again! The only time past performance influences a tournament is at Nationals, where you must qualify at a previous tournament. Otherwise, last week's performance means nothing when the new tournament starts!