As long as there's music, I'll keep on dancing.
Describing the world of Bayonetta in any way that makes sense would require a whole lot more familiarity and understanding of just what the heck is happening in that game than I posses, so I'm just going to say that this game is completely bananas, and leave it at what little I do know. You take the role of a ridiculous, over-sexualized witch named Bayonetta who uses her dark witchy powers to fight against a horde of insane looking monsters, which I believe mostly originate from Heaven. I don't have a clue what Bayonetta's goal in the game really is, or what any of the supporting cast has to do with pretty much anything. That said, this game is incredibly entertaining to experience and playing the game is nothing short of mind-blowing.
As a third person action-adventure game, many of the basics of Bayonetta are fairly common to other franchises - probably most notably things like Devil May Cry, God of War, and Ninja Gaiden. You have many combo attacks that generally work off of a light and heavy attack system, as well as guns you can blare at more distant targets. More crazy, cinematic attacks stem from the fact that Bayonetta's clothes are actually made of hair, and she can shape it into crazy demonic shapes to devour her enemies. Truth be told, every move Bayonetta makes and every word she says feels like a concerted effort to squeeze out every ounce of ridiculousness the developers could imagine, and it totally works.
Witch time is another key feature to combat, because it allows Bayonetta to effectively slow down time for everything except her, which of course lets her dish out some serious hurt in her unique way. This is activated by things like dodging an attack at the last second, and only lasts for moments at a time, but it makes the combat seem like it's in overdrive all the time, and rewards players who are a bit more daring.
Overall, Bayonetta's range of abilities and attacks is pretty impressive, and even ones that seem like they may be purely thematic attacks (like break dancing around the battle firing bullets rapid-fire in every direction) can be effective if used properly. You also upgrade your weapons as you progress, and by using different types of weapons you can change what combos are available, and some of the different weapons really have a different feel to them, which will change the way you fight.
The core action of this game is really just well done in every respect, and in some ways surpasses many of the big names that Bayonetta was probably inspired by. The nonsensical story and strange dialogue don't really detract from the game, because the unfiltered weirdness of it all somehow fits into a nice, cohesive package. But in an age of gaming where we can get huge well told stories with lots of cool characters on top of good gameplay, just doing one thing really well for a modest length game doesn't necessarily qualify it for greatness. Bayonetta does what it does so well that it is still in fact a wonderful game, it just isn't really for everybody, especially at a full $60.