The gaits used in ponyplay are the pretty much the same as the natural gaits of a horse. A horse has four basic gaits: walk, trot, canter, and gallop. In general, the goal is emulate a bio-equine's gaits. Certain gaits, such as the canter, are nearly impossible to truly reproduce; however, ponygirls and ponyboys can come very close and look good in the process (plus the training is lots of fun).
The walk is slowest and easiest gait to master. It is a four beat gait, which means that, in a bio-equine, one foot will always be raised off the ground at any given moment, while the other three remain on the ground. A four legged human pony can perfectly reproduce this gait by merely walking on all fours. For a two legged pony, this translates into simply walking as usual. Although a human pony's normal walk is a two beat gait, it can sound the same as a four beat gait (2 x 2 beats).
For two legged ponies, there are variants to the walk such as the high stepping and the "parade" walks. High stepping involves bringing the knee up higher than in normal walking, making a 90° angle with the ground. The pony's thigh should also be parallel to the ground. However, most of the time the standard walk is used since it is unrealistic to expect a pony to high step all the time.
The trot is a two beat gait. For a bio-equine this means that at any given moment two feet are in the air, while two feet remain on the ground. The feet move as diagonal pairs. The trot is best emulated by a four legged pony, but a two legged pony can trot by jogging at a moderate pace. The trot should be an energetic gait, but one that can be maintained for a reasonable length of time. At the trot, the pony should move freely and with good impulsion.
The canter is a three beat gait. When a bio-equine canters, one of his rear legs, for example the right rear, propels him forward. During this beat, the bio-equine is supported only on that single leg while the remaining three legs are moving forward. On the next beat the bio-equine catches himself on the left rear and right front legs while the other hind leg is still momentarily on the ground. On the third beat, the horse catches itself on the left front leg while the diagonal pair is momentarily still in contact with the ground. The more extended foreleg is matched by a slightly more extended hind leg on the same side. This is referred to as a "lead" (from Wikipedia).
This gait is a bit harder to emulate, but if you've ever seen a bio-equine at a canter you'll notice it can be reasonably well mimicked by stepping forward with one foot, meeting it with the other then repeating (albeit quickly). This is similar to skipping forward. This way your pony will have a lead at the canter just like a bio-equine. The pony's lead will be the foot he puts forward first, thus the pony should put the same foot forward every time (unless you're teaching your pony more advanced movements such as tempi, in which case the leading leg should be changed at the appropriate rate).
The gallop is very similar to the canter, but faster. The canter is a four beat gait; the only difference is that the second stage of the canter is split up into two stages so that the (in the example for the canter above) left rear would strike the ground, then the right front.
Since the gallop is the fastest gait of a horse, a ponygirl or ponyboy might choose to run full out when galloping. Obviously this will look quite different from a horse's gallop, but since the canter and gallop don't appear too dissimilar, it is useful to have a very fast gait.
I will try and post diagrams of the different gaits since it is hard to explain them succinctly; a good thing would be to watch a horse at each of its gaits and decide for yourself how best to reproduce each of the gaits.