They come in all different shapes, colours and sizes, I think mainly due to selective breeding though. But I only really know of quarter horses and quarter ponies, I don't know much about qhs, im from the uk. I have been to canda though and riden a qh, she was 15.1hh.
I don't believe AQHA recognizes different types per se.
Like iride said, there is the foundation type, which can also have a foundation registry. They are the typical QH build, very bulldoggy and muscular. And apprendix Quarter Horeses, which are first generation QH/TB. Those are the only AQHA recognized types. But it breaks down to stock and racing types.
Most horses that are used for pleasure riding and showing (english or western) are a mix of the two. With the exception of halter horses, which most prefer a stock type.
If you're asking about color, AQHA recocgnizes almost every color (more recently cremello) but only allows white on the legs and face, any more than that and the horse is unable to be registered.