Horses with extra movement patterns beyond the standard walk/trot/canter are called gaited horses.
Saddlebreds in the show ring often compete in “five-gaited” events – they can walk, trot, canter, rack, and slow-gait (many other gaited horses cannot trot at all).
Icelandic horses can perform several extra gaits, including the tölt (which is a lot like the rack or other ambling gaits) and the flying pace.
In the show ring, Paso Finos often display three separate forms of their classic ambling gait: the paso fino, the paso corto, and the paso largo.
Any horse can jump. Gaited horses are often used as trail horses. Any good trail horse should know how to jump with proper training.