in addition to the classes diamonr4ever mentioned, there is also western equitation. the rider is judged on their equitation and ability to ride and control their horse based on western equitation guidelines.
also, showmanship and, i believe, halter, can be shown in english attire. if you are showing in showmanship in english, it's basically the same except you're wearing english clothes and a helmet instead of a western hat, and your horse wears a bridle (english, of course) instead of a show halter. i know showmanship is like that; i'm not sure about halter, but i'm sure if would be fine. you'd probably have to check with the specific show's rules.
here is my heart and soul...please grind them into a hamburger and enjoy.