The name comes from the French verb "dresser," which literally means "to lift" and refers to the need of a ridden horse to lift its front end to compensate for the word of the rider. It is often, however, translated as "to train."
Dressage horses (except classical dressage, which is almost exclusively used purely for demonstration purposes these days) compete by doing a series of set movements called a dressage "test." There are different levels of test to show off the skill of horse and rider.
Four kinds of dressage exist:
1. Classical dressage, which is still performed by the Spanish Riding School, the French Cadre Noir, and a handful of other organizations. Classical dressage includes the "airs above the ground" - which I'll talk about in another post - and is based off of the training of a cavalry horse.
2. Modern dressage, which is an Olympic sport. The horse is ridden in the English style and performs tests which range from Introductory (where the horse is not even asked to canter) to Grand Prix, which includes maneuvers in which the horse goes sideways, trots in place and turns within its own length. Modern dressage includes freestyle (where the rider chooses the order of the movements and performs to music) and the pas de deux (where two horses and riders compete together in the arena).
3. Gaited dressage. Gaited dressage is a modified form of modern dressage which is designed for gaited horses. In gaited dressage, the horse performs some form of amble instead of the trot and the horse's way of going is judged by standards designed for gaited horses, who carry themselves differently.
4. Western dressage. Some western people have modified modern dressage for their own needs - which basically means judging horses by western standards of carriage and gait speed rather than English ones.