It's one of the "action-adventure" beat-em-up type deals in the vein of Devil May Cry, made for the PS3/360 and also made available through the Wii-U release of Bayonetta 2, I believe. On the subject of DMC, Hideki Kamiya was the director for this game as well. For better or for worse, he was also heavily involved with the story, which might explain why it's an incoherent mess held up on stilted one-liners and gratuitous anime-like violence. I know you might've played some of the DMC series, and those had some bare bones elements to them, but at least you could follow what was happening sometimes. Or the characters, despite being gimmicky in their own way, were put into relationships that were actually semi-believable so you could sort of invest yourself in what was happening while their stories played out. DMC 3, for example: Dante has the rivalry with his brother, and Lady wants revenge on her father for being an evil douche. Bayonetta? She's lady longlegs with guns and amnesia and several unspoken fetishes. And her character foils are: a chubby informant who is featured in, like, the intro and the credit cutscenes; a large token black weapon salesman/item vendor; a rival witch that is probably the only consistent relation with any depth, what little there is; a journalist that really wants to guilt Bayonetta and also be her love interest or something; a little girl who thinks of her as a mother, that's also used as a plot device so stupid that I still cannot believe it's in the story. Everyone Bayonetta interacts with is so detached from her for the majority of the game, and she doesn't seem to attempt showing empathy until almost the very end. And it's super awkward when it happens because 1) there's no sense of how she got to this point because it was never conveyed properly, and 2) whatever "development" that took place between the characters that were actually with her the majority of the time is nullified through storytelling things that really convolute the finished product and come off as silly because they don't contribute anything. I don't know. See for yourself if you happen to pick up the game.
The soundtrack's alright, but I wouldn't really call it a soundtrack. Jazzy, upbeat vocal tracks and orchestral flares for the occasional cutscene/boss encounter. But, that's all there is. This is a problem because the two primary fight encounter tracks are the same ones that you will be hearing whenever you fight mobs of enemies. With no variety or change until you get to those orchestral bits at boss time, during cutscenes, or in mini-game type sections. This is a problem because this a game based on fighting. This is a problem because while fighting, these two tracks are the only things you will be hearing over, and over. I really hope you like them (I do), because that's all the game gives you! Just a bad design choice in my opinion, especially when they showcased the talent they had at their disposal, only to do so little with it.
Visually, the game's great, as most contemporary titles are. Not much to say in that respect. It especially shows in combat, which I'll gush about in a little bit. Performance-wise, you might have to worry about some frame stuttering if you're playing on the PS3/360, but I expect more out of the Wii-U version. Level-design is mostly European cityscape-ish in influence, with the old, densely packed buildings being mixed in with modern amenities and architecture if that tickles your fancy parts. Also lots of floating platforms in empty space, but there's invisible walls so you don't fall off if you're fighting and not platforming, so it's k.
Now if Hideki drew a crowd for his storytelling and a few quirky design choices, it wouldn't be a very big crowd. No, he and his fans are in it for the gratuitous anime-like violence part that's done SO WELL that the rest of it doesn't matter! The gameplay is fluid, fast-paced, and as satisfying as any DMC I've played to date. Mobility is a key feature, giving you an effective double jump and dodge right from the start. This dodge also grants you the benefit of slowing down enemies momentarily to build your combos, if you can make a precisely-timed escape from an enemy's hit. (The amount of frames you're allowed is a little generous, but whatever. It's nice.) Combos feel as natural to execute as they should be when you're button mashing for your life/score. And if you want to get an enemy off the ground to deal with them more safely in the air, deal some extra damage, or just juggle, some of the combos reward you in these ways by, again, including timing as a requisite. Juggling enemies can be done, but at the risk of your health/score by potentially not noticing what other enemies are doing around you because you were too busy hammering Y and B. That's not entirely your fault though, because the camera is nothing to write home about. It has to fidget around pieces of the environment, or next to walls sometimes, but you can change its movement speed and the default invert settings (yes, people still do this for some reason) in one of the menus, so it becomes more manageable once you find a pace you're comfortable with. Otherwise, when you're just out in the open in combat, it stays at this 70-degree angle above Bayonetta and swivels around her position in a really zoomed out way. It'd be nice if it at least stayed consistent, but it has a tendency to focus in the direction of whatever enemy you've locked on to; more often than not this screws you over because attacking any enemy with ranged attacks tends to activate this effect, and it makes the camera not want to turn to see whatever bad guy's about to smack the back of your head. Or there are too many bad guys trying to smack all sides of your head and the camera decides it only wants to go a little bit higher than all of their heads and then stop, because overhead views are haaard. On a more positive note, the camera also changes for bosses and some mini-bosses, which is fine because they're pretty big, and it usually ends up in a Shadow of the Colossus kind of fighting style that can benefit from limited camera angles.