I don't know what brand they use in this vid but I bought one called a ConAir Chopper. It has the normal clipper head that has the fairly wide teeth that most clippers have. There is also a second clipper head included that is about 2/3 the width of the first and has much finer teeth. Since I installed the smaller head I haven't used the stock one at all. It works great for detail work on the goatee and if I did let what little hair I have left grow out for some reason the smaller head would give me a cut much like the one in the vid.
Right around $20 at Wally World.
I'm gonna do a little research to see if I can find out what brand the other one is, though. Maybe it's a pro model that barbers use?