No, your horse doesn't need to be registered within breed registries. With A/AA shows you may want/need to register your horse via one of the big organizations, like USEF. It's more or less a way to track the horse's performance in competitions, points they earn, etc.
And there *are* all sorts of breed shows, but the ones I'm talking about aren't breed specific at all. In fact, they're open to every single (non-gaited) breed out there. Everything from Drafts to Arabians.
Usually there's smaller hunter/jumper associations you'll see locally, then state associations. And the overall national organizations are USHJA and The United States Equestrian Federation ... Typically, they're all associated with each other.
It all just depends on your area. :)
Hmm thanks for the link, i'll have to google and see if theres any H/J associations near me. I think it would be really good information to watch some of those shows. That must of been fun kitten val lol, I personally wont be looking at riding green horses for english haha. I'm going to look at an experienced H/J tb, who was used for lessons. Older and experienced is more my style for learning something new on, I thought about trying to use my 4yr.old, but he just needs to much more work (really spooky), and I would probably be to nervous sitting on his back to focus enough on learning something :/