As explained by Wikipedia, the Free  Encyclopedia (www.en.wikipedia.org)
Dressage (a French term meaning "training") is a path and destination of competitive horse training, with competitions held at all levels from amateur to the Olympics.  Its fundamental purpose is to develop, through standardized progressive training methods, a horse's natural athletic ability and willingness to perform, thereby maximizing its potential as a riding horse. At the peak of a dressage horse's gymnastic development, it can smoothly respond to a skilled rider's minimal aids by performing the requested movement while remaining relaxed and appearing effortless. For this reason, dressage is occasionally referred to as "Horse Ballet."  Although the discipline has its roots in classical Greek horsemanship, mainly through the influence of Xenophon, dressage was first recognized as an important equestrian pursuit during the Renaissance in Western Europe. The great European riding masters of that period developed a sequential training system that has changed little since then and classical dressage is still considered the basis of trained modern dressage.

Early European aristocrats displayed their horses' training in equestrian pageants, but in modern dressage competition, successful training at the various levels is demonstrated through the performance of "tests," or prescribed series of movements within a standard arena. Judges evaluate each movement on the basis of an objective standard appropriate to the level of the test and assign each movement a score from zero to ten - zero being "not executed" and 10 being "excellent." A score of 9 (or "very good") is considered a particularly high mark, while a competitor achieving all 6s (or 60% overall) should be considering moving on to the next level.

There is nothing more breathtaking than watching dressage Olympic competition, as these horses move and "dance" in a stylized manner. Expert dressage riders' cues to their mounts should be imperceptible. The team appears to be working in a partnership reminiscent of highly skilled dancers performing a pas de deux in ballet, with two dancers moving effortlessly in rhythm, choreographing a brilliantly flowing motion.  As the dressage horse and rider become more proficient in training, their levels of competition are more exacting. Gait changes, speed and collection that appears to emanate by imperceptible communication are the hallmarks of a great partnership between horse and rider. It takes years of exacting training and hard work to achieve this. Many great dressage horses appear to enjoy performing and competition.

There are several levels of dressage, each more difficult as the levels progress, and two forms, classical and competitive.   Classical dressage requires a much greater and more strenuous effort on the part of the horse. Horses are asked to perform "aires above the ground," which are not a natural part of the horse's movements. Rooted in military arts, where horses were trained to attack adversaries by kicking, striking out and rearing maneuvers, this aspect of dressage is one for which the Lippizaner Stallions are famous.     Competitive dressage, on the other hand, is based on the natural moves that are part of horses' gaits such as the Piaffe, Passage, Hall-pass, Extended Trot and Tempi Changes.   You can view some of them in these Wikipedia videos:   the piaffe, passage, half-pass, extended trot, pirouette and tempi changes.

USEF INFORMATION ABOUT DRESSAGE
The United States Equestrian Federation (USEF) established six official levels of competition standardized throughout the United States: Training, First, Second, Third, and Fourth. Training is the level of least difficulty and Fourth is the level having more difficulty. The International Equestrian Federation (FEI) established four additional levels of competition which are standardized throughout the world: Prix St. George, Intermediare I, Intermediare II and Grand Prix, with Grand Prix being the highest level of all Dressage Competition.  Each level has appropriately, pre-choreographed 'tests' which competitors perform, individually, in front of a highly qualified judge. As each movement of the test is executed, it is numerically rated from 0 to 10, 10 being the highest possible score for any given movement. All the scores are then added, assigned a percentage rating and compared to other competitors scores in the same test and level.

THE DRESSAGE ARENA(S) 
There are two sizes of arenas: small and standard. The small arena is 20 meters by 40 meters, and is used for the lower levels of dressage and 3-day eventing dressage. The standard arena is 20 meters by 60 meters, and is used for upper-level tests. Dressage arenas have a lettering system around their outside in the order (clockwise) A-K-E-H-C-M-B-F (small arena) and A-K-V-E-S-H-C-M-R-B-P-F (standard arena). At the start of the test, the horse enters at A. There is always a judge sitting at C (although for upper-level competition, there is generally more than one judge at a second or third place around the arena, and in some competitions there can be 5 judges located around the arena). The invisible letter X is always in the center of the dressage arena.  The dressage arena also has a centerline (from A to C, going trough X in the middle), as well as two quarter-lines (halfway between the centerline and long sides of each arena).