The fundamental purpose of Dressage is to develop, through progressive training methods, a horse's natural athletic ability and willingness to perform, thereby maximizing its potential as a riding horse.  All riding horses can benefit from use of dressage principles and training techniques.  Dressage is an egalitarian sport in which all breeds are given an opportunity to successfully compete.

Glossary of Dressage Terms

Acceptance of the Bit--The horse accepts contact and commands from the bit without fighting.

Aids--The ongoing non-verbal communication of the rider to the horse using the legs, seat, and reins.  May also include use of riding crop and spurs.

Canter--The canter is a controlled, three-beat gait that usually is a bit faster than the average trot, but slower than the gallop. Listening to a horse canter, one can usually hear the three beats as though a drum had been struck three times in succession. Then there is a rest, and immediately afterwards the three-beat occurs again. The faster the horse is moving, the longer the suspension time between the three beats.

Collection--At the apex of the training scale stands collection. It may refer to colleted gaits: they can be used occasionally to supplement less vigorous work. It involves difficult movements, such as flying changes) in more advanced horses. Collection requires greater muscular strength, so must be advanced upon slowly. When in collected gait, the stride length should shorten, and the stride should increase in energy and activity.

Contact--Contact is the result of the horse’s pushing power, and should never be achieved by the pulling of the rider’s hands. The rider drives the horse into soft hands that allow the horse to come up into the bridle, and should always follow the natural motion of the animal’s head. The horse should have equal contact in both reins.

Free walk--Relaxed walk in which the horse is allowed to lengthen and stretch to the bit.

Gait--Horse gaits are the different ways in which a horse can move, either naturally or as a result of specialized training.
 
Half-pass--Lateral movement in which the horse is bent in the direction of the motion.

Impulsion--Horse's desire to move forward, including elasticity of steps and roundness.  However, speed does not create impulsion, and a rushing horse is more likely to be "flat" than impulsive. A horse can only have impulsion if his hind legs are engaged and his elastic back allows the power to come through.

Lateral movement--Any movement which involves motion in a sideways direction.

Lead (canter)--The foot which strikes most forward in the canter stride.

Leg yield--The leg-yield is a lateral movement performed on a horse, in which the horse travels both sideways and forward at the same time. The horse is fairly straight through his body in the leg-yield, although he may have a slight bend to the outside (away from the direction of travel).

Medium walk--Normal walk in which the horse is kept on the bit and in a correct frame for his current level of training.

Posting--A way of riding the trot in which the rider sits and rises in rhythm to the horse's trot strides.

Riding Techniques--Wikipedia has a quick reference for various riding technique terms.

Relaxation--Signs of looseness in the horse may be seen by an even stride that is swinging through the back and causing the tail to swing like a pendulum, looseness at the poll, a soft chewing of the bit, and a relaxed blowing through the nose. The horse will make smooth transitions, be easy to position from side to side, and will willingly reach down into the contact as the reins are lengthened.

Roundness--The way in which the horse travels with his back "lifted" into the saddle rather than hollow.

Submission--The horse's attention and confidence, harmony with rider, lightness of movements, and acceptance of the bit.

Trot--In this gait, the horse moves its legs in unison in diagonal pairs. From the standpoint of the balance of the horse, this is a very stable gait, and the horse need not make major balancing motions with its head and neck. The trot may be ridden sitting or rising/posting. 

Walk--The walk is a four-beat gait.  When walking, a horse's legs follow this sequence: left hind leg, left front leg, right hind leg, right front leg, in a regular 1-2-3-4 beat. At the walk, the horse will always have one foot raised and the other three feet on the ground, save for a brief moment when weight is being transferred from one foot to another. A horse moves its head and neck in a slight up and down motion that helps maintain balance.

“Be clear, be patient, and repeat.  Sit still.”

A standard sized arena is 20 m by 60 m.  The standard dressage arena letters are A-K-E-H-C-M-B-F. The letter X in the middle ia not actually marked, but is indicative of the center of the arena.

The dressage arena also has a centerline (from A to C, going through X in the middle), as well as two quarter-lines (halfway between the centerline and long sides of each arena).