There's straight Swiss. All players play the same number of rounds. Each round players are paired against opponents that have the same win-loss record. At the end, you should have one undefeated player, who is the winner. This is most commonly used for pre-releases.
There's Single Elimination. I think that's obvious. It's like all the college basketball tournaments -- lose and go home. No "official" event uses straight single-elim, but some local tournaments might.
The most common format in Pokemon -- Swiss plus Single Elimination -- combines the two. The tournament begins with all players playing Swiss rounds. At the end of Swiss, the top players are seeded into a Single Elimination tournament. This is the format used for all Championship Series events (Battle Road, Cities, States, Regionals, Nationals, Worlds).