I'm sure that this thread will have LOTS of posts, but I saw it first, so here goes...
DRESSAGE is gymnastic horse training. I suppose it began several thousand years ago to train a horse to warfare, but the first surviving writing is from the Greek, Xenophon, who wrote about kind but correct horse training. (He also wrote on agriculture and other topics, too.) Jump to the 1300's in Europe and several schools specializing in military horse training sprang up. The most famous TODAY to the general public is the Spanish School of Riding in Vienna. That is probably bc General Patton helped to save the Lippizaners from becoming meat for the Nazi army and most people know something about WWII. They are other schools in Europe equally well known, but mostly to Dressage enthusiasts.
Commonly horses are trained for impulsion (to move readily forward, and moving backwards is considered a forward movement bc of the footfalls). They are trained to flex their necks in order to balance the whole body. They are trained to be light on the bit and flexible in the necks and through the ribcage, in order to be soft and responsive to the rider's cues. The rider sits deep and "collects" and "extends" the horse's gaits. A great deal of the horse's training is to develop the musculature to compress and spring through the walk, trot, and canter. Some of the training is developed outside. Col. Podjawsky recommended outdoor cross country riding and jumping to help muscle up his horses. Mostly, training takes place in the arena.
A good place to research is the USET (United States Equestrian Team.) United States Equestrian Team Foundation
You should definitely watch this year's summer Olympics Dressage competitions. They will be in England, and you won't need a translator!