Dressage- All done on the flat, in a rectangular arena. The "patterns", which are called "tests", stay the same for four years, so you go to a show knowing exactly what you need to do, and you'll be judged based on how well you do it. There are letters placed around the arena, and you're supposed to ride very accurately, so that each transition is done right next to the letter. Both you and the horse are judged, but I would say most of the attention is on the horse.
Jumpers- You get to the show, where you will be told the course (jumping pattern). The jumps are usually brightly colored. The goal is to jump the course as fast as you can (while remaining safe of course), and the fasted horse/rider wins. Judging isn't subjective, it's just based on time. Clothing/tack isn't as strict as...
Hunter over fences- You get to the show, where you will be told the course (jumping pattern). The jumps are a little more natural looking. Your horse is being judged on how well he jumps. You don't need to worry about time, just jumping well. Tack/clothing is more conservative and traditional.
Equitation over fences- Same as Hunters, but YOU'RE being judged on your jumping ability, not the horse.
Eventing- 3 phases. Dressage, jumping, and cross country. Cross country you are out in the open (not in an arena), jumping solid "natural" obsticals (including water jumps), and you have to fit in to a time limit, so you're doing it fairly fast. Sounds scary, but they have lower levels to get you started and they're fun. The jumping phase is more like show jumping, which I believe is closest to jumpers, but someone else may have to clear that up a bit, because I'm not sure.
I hope this helps! To other Horse Forum members, feel free to add to this/edit it if you need to.