Yes, horses do most of their sleeping standing up. Their legs can lock in place, enabling them to fall asleep without falling over. Because they are prey animals, horses often don't feel comfortable sleeping on the ground, and most of their sleeping is done during the day rather than at night when the predators are out hunting.
Horses have straight backs, so they cannot get up quickly. If a predator were to come while a horse was on the ground, they might not be able to get up fast enough to get away. However, horses do occasionally take short naps laying down during the day. This helps them to rest their legs.

It's believed that in order to get a deep, REM sleep, they need to lay down. You can sometimes find a horse stretched out on its side, asleep in the sun, or laying on the ground with its legs folded under. When horses are in groups, they will often take turns 'guarding' each other as they rest, with one horse standing up near the sleeping horse. This behavior would help to keep a wild horse from being attacked during a nap.