What's the name for "horse dancing?"?

I think you are referring to dressage - that is what the rider in the video is doing. Dressage actually means training (the rider is training the horse to hold himself and use his muscles properly). But, it does look like dance. The video is of a freestyle competition. In a freestyle program, the user can design it any way he/she likes and usually does it to music. Most dressage competitions though have tests. They start with Training Level Tests (they even have a Walk, Trot one). The tests tell you A,exactly what to do at the different letters in the dressage ring (the letters are A,K,E,H,C,M,B,F). So, a training level test (one I've done a zillion times) will start like this: Enter ring, turn at C (so you go up to the middle of the ring), Halt at X (X is between H and M, if I remember correctly), C - track left, 20 meter circle at E, between E and K, canter transition, etc. Certainly not the dancing that higher levels provide but the lower level tests are meant for both rider and horse to improve. Each movement gets graded from 1-10 (10 is the best). The tests get harder and harder until you get to Grand Prix. If you can, I would definitely recommend taking some dressage lessons. Even if you are going to be a hunter/jumper rider, it helps to learn how the horse should be using himself. Hope that helped.

dressage. maybe you are talking about freestyle dressage