Note that each competition should specific rules. If you're thinking about competing, READ THE RULES! If you just want an overview of what's out there, this page may help you understand videos you watch.
Jack and Jill
This format is design to test your ability to dance socially. To do this, you get paired with a random partner and dance. Often you will get three songs with three different partners and may get different tempos or styles with each partner. Depending on the level the format may be All-Skate or Jam Session style.
Strictly _____
In a Strictly Lindy competition, you must dance only Lindy. You get to choose your parner and dance with many other people (at least at first). You don't know what the song is, so you can't choreograph anything.
Classic
In this type of competition, you can choose your partner and the song. Each couple goes one at a time before the entire audience.
Showcase
This is just like classic, except it's common for dancers to push the boundaries of the dance. Arials would be more accepted here.
Styles
In an All-Skate, everyone (or maybe on "heat", commonly half of everyone) will dance all at the same time. In a Jam Session, each couple goes out one at a time as the song progresses and each couple typically gets "eight eights", or 16 measures. It's common for swing music to be phrased in 16 measure groups.

