Also known as the Battle Dance, the Victory Dance and sometimes the Cossac Dance --though I honestly have no idea why about the latter. The Bionic Dance is a classic componant of almost any anime, usually right before a major battle. It is also where a lot of inspiration for the incredible camera effects in The Matrix. It only does justice if you see it, but here's an attempt at description.
Generally the protagonist and an antagonist are right about to fight. After one of them cries out how they will defeat the other they begin a series of katas, while the viewpoint (or camera angle) rotates around the character, zooming in and out. The background itself is usually reduced to flat black with a few slashes of color or action lines or possibly even a battle aura.
During this dance they generally do one of two things: call upon the source of their powers (such as Voltron invoking his sword) or they announce their no doubt rare and unique attack form.
After this they usually launch some sort of power attack against their enemy... or, in my favorite use of the Bionic Dance, Captain (then General) Justy Ueki Tylor uses the Bionic Dance, silently, to salute his opposing General in The Irresponsible Captain Tylor.
Since it's inception it has grown to be amazingly popular. Some of the first anime to use it were Cutey Honey, Voltron, and Thundercats. The cult film AKIRA even uses it at one point. Now you can find it regularly in American cartoons as well, such as The Powerpuff Girls, and Dexter's Laboratory. I've even seen a very disturbing muppet version of it on Sesame Street.