Rule of Reason pretty much nailed it. Well boiled down, Hunters focus on the horse, equitation on the rider, and jumpers on the clock and the course.
Hunters are judged according to their way of going, manners, and style of jumping. A good hunter round looks very smooth and relaxed, with the rider in complete control while still allowing the horse to work. There is a very distinct hunter "look" in a horse, usually Thoroughbred-y in type, with long and clean legs, and a flat, ground covering movement on the flat and a scopey jump. In attitude, a hunter needs to be willing and obedient, but also convey the endurance that would be necessary to take a day in the field, as he would actually riding to hounds. Hunter classes still tend to be conservative and minimalist in turnout - brown leather, a white fleece shaped saddle pad, snaffle-type bit with plain cavesson bridle, beige breeches, and black/dark grey/navy/hunter green jacket. Jumps are usually natural looking, either plain wood or white paint with brush or flowers, that sort of thing.
Equitation visually looks similar to hunters, and a lot of riders will compete in both divisions. The focus is, of course, on the rider's position and skill in guiding the horse around the course, so there isn't so much emphasis on the horse's type, conformation, or gaits, although outright misbehavior or unsafe jumping won't earn you any points.
Jumpers is essentially like European show jumping or stadium jumping. The object is to get around the course as fast as possible with minimum faults - there are no "style points" in Jumpers, and the ideal horse is one that is "handy;" take the course clear, and make turns tight enough to shave fractions of seconds from the round. I like to think of Jumpers as the English world's rough equivalent to barrel racing, in terms of the object of the game, and, very generally, specialized Jumper horses tend to be a bit higher-octane than Hunters. In my understanding, tack requirements are more lenient in Jumpers than in the other divisions, allowing martingales and stronger bits. Visually, jumper classes look a bit more flamboyant than hunters or eq, especially in youth divisions. Square pads in colors are more common, as well as bright boots/bandages and shirts. Fences are generally more fantastic, painted bright colors with more unnatural decorations and "spookies," like flags, etc.