A liberty horse is a show horse that is trained to perform without a harness or other tactile input; they can complete a routine given only verbal commands, visual signals, and/or whips -- although the whips are visual and auditory signals, and do not actually touch the horse. This usually means that they have no rider, although at times a human may sit or stand on their back as part of an act. They are generally found at dressage exhibitions or in circuses, and may be most familiar to most of us due to their appearance in Cavalia and more traditional trick riding circus acts.
In circuses liberty horses are most often trained to move in formations, such as wheeling, circling, running in file, or even jumping though rings of fire. However, there are plenty of solo liberty horse performances, both at circuses and at dressage shows.
In the world of liberty horses, 'liberty' is used to refer to the horse itself, the style of training used, the communication between the horse and the trainer, and events/acts using liberty horses. To add to the confusion, liberty training is a separate although related idea, referring a system of training that does not use bits, bridles, and spurs, but does not usually mean that a horse is being trained to do tricks.