So, here is part 1, straight from Wikipedia ...
Riding figures are figures performed in a riding arena, usually for training purposes. Figures may also be performed out in a field or other open area, but a riding arena provides markers that can help indicate the correctness in the size or shape of a figure.
Figures are required components of dressage tests, are used in reining competition, and may also be asked for in equitation classes. Additionally, jumping courses may often be broken up into riding figures.
It is important to work the horse on figures in both directions, to ensure an equal build of muscle on either side.
End of part 1 ... more to come.