Gymnastic exercises are a series of ground poles and jumps with no more than two strides between that are commonly used in jump training for both horses and riders. They can be set up a million different ways: different number of jumps, ground lines, verticals, oxers, bounces to different degrees of difficulty.
Here's a really great overview and sample of an exercise from Nona Garson, an Olympic veteran: http://www.youtube.com/watch?v=_dqAk9nKlxc
When setting up gymnastics, it's important to understand the stride distance for setting jumps. The average canter stride is 12', and you need to allow 6' for landing and 6' for take off. So, to set a jump with one stride inbetween, there should be 24' between the jumps (6 for landing, 12 for the stride, and 6 for the takeoff). A bounce (no strides) would be 12'. When doing gymnastics, this is a good place to start, and then you can adjust as necessary for a horse with shorter or longer strides. It's important to make the jumps a comfortable distance for the horse so he can go right through without having to adjust.
One of my favorite gymnastics is a ground line, a cross rail, one stride to a vertical, and a bounce (three total jumps). Start with the cross rail with a ground rail half a stride in front. Trot in and canter out (typical start for a gymnastic). Once the horse is comfortable, add a vertical one stride (24') beyond the crossrail. Keep it low (2' ish) to start. Once he's comfortable, add some height to the crossrail. Once he's comfortable with that, add a third jump with no strides in between (12'). Keep it low to start, and then work the height up as he's comfortable. The bounce in particular is great for getting him to round his back and neck. It's key to build it up slowly so as not to scare the horse or make him uncomfortable. Always trot in and canter out. The rider should go to 2-point over the ground rail and hold it until she's all the way through the gymnastic. The idea is to feel the horse coming up to her over the jump. Nona explains this really well in that video.
An easy way to set the second and third jumps for a gymnastic (if you have someone helping on the ground) is to place a ground pole where the jumps would be (ground line, cross rail, two ground lines). This allows you to watch as the horse trots in and canters out and set the poles to match his stride before adding any height.
The gymnastic in Nona's video is a great one to start with. It's a ground line to a crossrail, one stride to a vertical, one stride to an oxer. This is a great one for a first go around. Remember to build it up slowly, though, one obstacle at a time.
Gymnastics are very very very useful exercises for both the horse and the rider, and they are a whole lot of fun. Oftentimes, we'd set up a gymnastic on the long side of the ring and use it to start a course of jumps. It helps get the horse's impulsion up and gets him focused.
Hopefully this all makes sense; I've done gymnastics a million times and done it with students, but never tried to type out an explanation. Let me know if you have any questions ... other posters add your favorite gymnastic exercises!