oh man, big question
Rec Ball:
This is where you and some random people split into two groups and go play lots of brief games, usually in the woods, with maybe some games on speedball fields. The speedball fields could be anything from hyperball to ultimate air to crates, tires, and barrels. No score is kept. The emphasis is on having a good time. Everyone tells silly little stories about the best and worst games they had. There could be a huge mix of skill levels, including never-played-befores and frustrated-ex-tournament-taking-out-rage-on-punters. The latter end of the spectrum is frowned upon, and rather sad.
Scenario:
This is a little more organized, but still basically good times goofin' around. A much larger group will get together at a date announced well in advance. It will be split into two (or possibly more) sides, and there will be a story structure to one long game that may last several hours to two days. Points are awarded for a variety of objectives; each side will have a general or boss or whatever who tries to maintain something like awareness of what is going on and some small degree of control of his team. At the end, there is often a big drawing and cool stuff and some lame stuff will be given out at random. There may also be awards for MVP or sportsmanship or whatever, usually by each general.
Tournaments:
This is where people start getting serious about winning and keeping score. Tournaments are almost always played in a speedball arena, and as time goes by, there are fewer and fewer barrel, tire and crate fields in use, and less hyperball fields, and more Ultimate Air style fields. There are local leagues, which are often based on 5 man team competition. There are Majors, too- the NPPL is an American League with 7 man teams, the PSP has Xball and 5 man (if 5 man hasn't disappeared yet- I forget), and the MIllenium Series, which serves Europe with 7 man competition.
Xball:
This is a tournament variant where two teams field five players repeated against one another for one point each time they take the field. There is a clock, and the team that scores the most points in the game time wins. Penalties are assessed for certain acts of misbehaviour. Teams may have many people involved, including multiple lines of players, coaches, gun techs, and cheerleaders (just kidding about the cheerleaders). Organization and quick action in the team pits is needed for success in this format, much more so than in other formats. Temas are less likely to prevail due to luck.