Velvet gave a good basic explanation.
Hunter classes are supposed to be ridden forward seat, in conservative tack and clothing, and judged primarily on the horses suitability to be a field hunter. The fact that 95% of the show hunters would have a case of the vapors and collapse if faced with actual hunting conditions is somewhat beside the point.
Hunters should be judged on manners, movement, way of going, soundness, substance and style. The should be shown on a light, passive contact with an easy, forward, ground-covering stride.
Where it gets *really* confusing is when you add in the breed show definitions and then gaited horse definitions.
"Hunter" and "Hunter under Saddle" under hunter or USEF rules is a completely different thing then QH, Arab or Morgan "Hunter Under Saddle." Breed show "hunter" classes are more about the breed standard and showing the breed's versatitlity and less about the hunter standard. They show mostly in similiar tack and clothing but have very differents ways of going.
Then you have gaited show classes like "English Pleasure" and "Country Pleasure"; you would need a gaited person to tell you about those.
Dressage is about the systematic gymnastic development of the horse as an athlete. Dressage competition is the performance of a specific test or tests with defined movements performed alone in a standard-sized arena with lettered markers. Dressage riders ride with a longer stirrup and deeper seat than forward seat riders, and ride most gaits sitting. Dressage emphasizes not just a systematic training approach to developing the horse's musculature and carrying power, but also the horse's submission and obedience to the aids.
My non-horsey brother once described it as "the horses doing that airy-fairy stuff with their feet." For many non-dressage riders, that's a fairly sufficient explanation.