Also known as "dog dancing", but not if you're a professional.
It's been around for about 15 years, has several organizations devoted to it, and you too can join if your dog will fill your dance card for you. (Did you see that episode of "King of the Hill" too?)
Canine freestyle involves training, communication and choreography between the human and the dog. You can get an inkling of what is involved when you visit, say, the Canine Freestyle Federation. You can find classes, get training, even go to the retreat held every two years.
A couple of years ago Emily Yoffe at Slate tried the sport with her dog.
And you can see some on YouTube.