Courtesy of tumblr user norvicensiandoran.
Horses do not always have at least one hoof on the ground when moving or running (and obviously not when jumping).
At the walk, yes, a horse always has at least one hoof on the ground, but the other gaits - trot, canter, and gallop all contain what we call the "moment of suspension" - a point in the stride when all four hooves are off the ground.
The speed of a racing trot shows the moment of suspension clearly (the back hoof is a bit hard to see as its between the sulky wheels. (Image source: Wikimedia Commons).
This appaloosa is cantering and again caught in the "moment of suspension."
The existence of the moment of suspension of the trot was first documented in 1872 (when photography started to become more useful). Until then, it was only suspected.
The pace also has a moment of suspension. However, in all of the variants on the running walk, amble or foxtrot, there is no moment of suspension - which is part of why gaited horses are so comfortable over long distances.