Most horses have four gaits that they commonly use. The first, the walk, is a slow, tame, four beat gait. The trot, slightly faster, is a two beat gait. The canter, a three beat gait, has a rocking feel to it. Last but not least is the gallop, which is a fast four beat gait, averaging about 25 mph.
The most comfortable gait for a horse to jump from is the canter. This is a consistent gait that allows the horse enough momentum as well as an even pacing so that an adequate amount of force can be used during the takeoff.
My trainer has told me that there are five components to a jump, all important in making sure that you complete it safely and well, gracefully.
The first is the approach. Approaching the jump means that you must prepare at the correct angle, make sure you have enough room to count out the proper stride length, and give yourself time to think about whether you need the horse to go a little faster (a small nudge with your calves), or slower (a half-halt, pulling slightly with both reins and then releasing), in order to glide into phase two safely.
Phase two is the actual jump itself. When jumping, the horse exerts a force on the ground in order to push itself upwards. This can be represented by Newton's second law or F=ma. Horses can carry up to 30% of their body weight. While this is impressive, it is also a limit. When the horse exerts all of that force on the ground, the ground exerts the same force back. This can seriously damage their back legs if the rider is not careful.
Phase three of the jump is the air time. While the horse is in the air, it is the rider's job to not only sit up, releasing the pressure on their back, but make sure that their landing will be comfortable and set them up correctly to continue the course. During their time in the air, the horse has only potential energy, compared to it's kinetic energy during the approach. At the horse's maximum height, the velocity is zero, meaning the only force acting upon horse and rider is gravity: 9.81 m/s^2.
Phase four is probably the most dangerous phase of the jump, not only because there is a large force between the horse's front legs and the ground, but also due to the fact that all of the rider's weight is put onto the horse's front legs as well, causing a large strain. Again, if the rider does not properly care for the horse, all of this weight and force could potentially damage it's legs.
Phase five, or finale, is after the jump is completed. The velocity of the horse should return to that of the velocity during the approach, meaning the sum of the forces during the entire jump should be equal to zero. The rider should check to make sure the horse is on the correct lead, and proceed to evaluate their next jump, repeating step 1.