Here in the States there's 3 main types of classes held at "Hunter/Jumper" shows.
Jumper classes are your basic stadium jumping classes.
Hunter classes are judged on the horses' smoothness, consistency, form over fences, way of going, and in some classes their conformation is judged. The rider is not penalized unless they do something to hinder the horse. Originally, horses showing in hunter classes should be of good build, with a long, smooth stride and were supposed to represent the ideal field hunter and was 'to be shown over a course of fences built to represent those found on the hunt course'. But of course, we Americans have to take everything to a whole other level and nowadays it's more or less about who has the most expensive pony and the jumps in the ring look NOTHING like what you'd find while hunting. And most of the horses and ponies have exaggerated gaits and all trot around heavy on the forehand. Thank goodness though, there seems to be a revival of the "real" show hunters brewing...
Equitation classes on the other hand are judged on the rider. Riders have to show off their form (body, hand position, leg position, etc), and their skills at guiding their horse over a course of fences. In the lower levels its just a small course of cross rails, but the "Big Eq" and Adult Equitation classes are much more difficult, with various tests that can be included.
What sort of classes do they have where you are? English horse shows here are very different!