What type of dancing is this?

That type of dancing isn't really done anymore, it's kind of old. The closest thing to it would probably be the Australian New Vogue style, or the English Old Tyme style. Both are forms of ballroom dancing. They're also often called "sequence dancing", because everyone does the same steps, so it is easy to switch partners, i.e. you dance in a circle and at a certain point in the dance the lady will move onto the next partner while the gentleman stays in place.