It is difficult to write an article on the the leg aids for a horse because many horses are traineds slightly differently. There are several common leg aids for a horse, but keep in mind that the common leg aids, and the way a particular horse understands them, may be affected by how the horse was trained.
The basic leg aid is pressure on the horses sides along with seat cues to cue the horse to transition upward or speed up. Legs taken off the horse's side usually tell the horse, along with other cues, to slow down. To signal for a canter or lope on most horses, slide the leg that is on the outside of the horse's bend slightly back behind the girth and apply pressure. This is the leg aid to ask for canter. Some horses know more advanced leg aids, such as how to leg yield or side pass. Leg aids are only part of the cues to make a horse side pass or led yield, but the leg aids are typically pressure on one side and no pressure on the other. This signals a horse to step away from pressure in to the open leg.
These are the basic leg aids for most horses. Very green horses will know less leg aids, and very advanced horses will know many more leg aids. If you are confused about how to use leg aids, or how your horse understands leg aids, you should consult a professional instructor.