Answer:
It depends on the type of horse you are talking about. A normal, everyday sort of breed then they would have 4 gaits, walk trot canter and gallop.
Some breeds have more than four gaits. For example, the Standardbred can pace, the Tennessee Walker has a running walk, etc. This gaits are unique to their specific breeds.
Within each gait, there are variations, especially for a dressage horse. These are extended, collected, working, and medium. However, these are variations and NOT gaits. A gait is defined by a unique footfall. Variations, on the other hand, take place within a gait and have to do with the speed of the gait and the collection of the horse.