Riders sit on top of the rib cage, which is an elastic structure by virtue of its connections with the spine and the amount of cartilage connecting it to the sternum. It is remarkable for several reasons:
1) there are no floating ribs (connected just at the spine): ribs all are connected by cartilage at bottom of the trunk,
2) the horse's center of mass is located within it,
3) it is adjustable in term of its "bounce response" or "tuned reaction" by means of neck and pelvis position acting via elastic ligament or tendinous tissues, and
4) the ribcage may act as a "spring-loaded" mediator of bend in the thoracic spine.
These features create a structure which might be compressed by the grounded leg (potential energy) and could immediately make it available to the airborne leg (kinetic energy). Presumably, a horse can adjust its spinal posture according to what it needs to do: stand, walk, trot, canter (maneuverable gait) or gallop like hell (optimal covering of ground). Millions of years of dealing with predators have honed equine conformation to be excellent for middle distance running: lead changes (change of bend to the rested diagonal pair) allow the other set of muscles to extend endurance, not to mention the ability to kick while moving.
But we want to ride this animal. As La Guérinière has pointed out, we will not want to ride everything the horse can do!
Adjustable bounciness is a key to understanding aids which a rider gives. It is simple to state what a rider may do: doing it is very difficult.
Basically, a rider is able, by using lower body aids, to adjust the posture of the horse by positioning its spine so the rib cage is set up for each movement. Half halts are the name given to the aids by which a rider asks the horse to adjust his posture in order to control the joined centers of mass. As you can see from the second diagram below, these should should come mainly from the lower body, which is placed so that the rider may affect crucial muscles which "tune" the rib cage. If that were all we had to know about dressage, we could all go home after reading the above sentences.
Because the postures a horse can achieve lie along a continuum, there are a huge number of positions possible. A rider must learn them by feel in order to give an appropriate half halt. That is why a ground person with sharp eyes and a sense of what is most important at a given moment is essential to give the rider feedback on what is functioning correctly. It is up to the rider to inform the ground person how the movement feels.
The horses in the images below are in a posture which allows them to be "on the contact" rather than "on the bit." Distinguishing the feel of these two conditions is essential in moving a horse from its basic gymnastic schooling to middle and high schools of athletic development.