I got asked about this on Facebook.
Horse dancing is a relatively recent tradition that exists in south and central America, especially in Mexico. It's similar to the dressage movement the piaffe - the horse is trained to trot in spot. The difference is that these horses do the movement for an extended period of time, making it a test of stamina.
Horse dancing has a very bad reputation - it's often considered cruel. The reason is that many of these trainers try to use pillar training (I'll go into more detail about this tomorrow), which is a classical dressage training method that is effective if used correctly...and inhumane if done wrong. (Rather like double bridles and spade bits). And, of course, some of these horses forget how to move forward freely.
Dancing in and of itself, though, is not cruel and can be done in a way that's natural for the horse (in fact, the piaffe is based off of movements stallions use to demonstrate their prowess in the wild and many horses, especially male ones, will do it at liberty).