osu! is an excellent rhythm game with lots of beatmaps available (though of varying quality), as well as a fantastic online mode and a couple of special modes that help to add some variety (Taiko, Catch the Beat). Probably the best free rhythm game. Video of what some people have done with it
here - beatmaps are crazy customizable, what with skins, backgrounds (with storyboard animation!), and sounds all allowed to be overridden by a specific song. It really helps to make beatmaps feel different from each other. Some of the storyboarding can even change based on if you're doing well or poorly.
It
is kind of meant for a touchscreen, but I manage to play pretty well with the mouse. The other modes are designed for a keyboard, though, and are generated from the same beatmaps as regular songs. If you find it too difficult, I recommend the autopilot mode (where you just click and the mouse moves on its own).
I also have had an incredible time with
DJ Max Trilogy, which is a much more traditional rhythm game. Press the keys (four to eight) in time to the music, standard stuff. It differs from most rhythm games in two big ways: for one, it's designed with a keyboard in mind, and two, instead of key presses just letting a prerecorded background track keep playing, they actually generate a sound (though background music usually also plays).
For example, all the sounds in this video are from the player pressing keys; no background music plays (by choice). All the songs are excellently mapped, and there are over 130 tracks counting updates, all playable with 5 possible key modes (4, 5, 6, 7, or 8 rows of notes) and multiple difficulties for most key modes, as well as an online multiplayer mode, mission mode, and stage mode (think DDR). There's an insane amount of content here and it's totally worth it if you're really into music games. Unfortunately, the only way to get a hold of it that I know of is to import it which cost me $70. I still think it was worth it though, I've gotten a ton of playtime out of it. It's the best music game I've played, I think.
Also! In slightly more practical news,
Rocksmith is due out soon. I'm interested in it because it's a music game where you plug in an actual guitar and play actual songs with it. It's also getting a PC release instead of being console exclusive! I'm eager to play it.
I've spent way too much time playing music games. 
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