They often specialize in Hit-and-Run Tactics, especially with archers who can spur their horse, turn around and shoot backwards as their enemies pursue. The first category are the light horse archers, who act as skirmishers and rely on agility and speed. There are basically two categories of this trope. The Huns and Mongols are the best-known of these, and have spawned any number of clones in fantasy literature, but almost every civilization that had horses has used these at some point, often to devastating effectiveness, and the Samurai used the dai-kyu (a type of recurved longbow) from horseback up until the Meiji Restoration.
Prior to the invention of the stirrup, lance warfare demanded a great deal more skill as it required both hands holding the weapon. Truth in Television for numerous civilizations, of course, with many historical armies being made up partly or primarily of horse archers. Someone who shoots a bow while on horseback.