Sounds like a generic WW2 game. I'll stab in the dark and guess Call of Duty: Black Ops.

It's actually not. May Sound like some kind of ww but it isn't. I'll give you a clue, UMBRA WITCHES and the LUMEN SAGES fought against each other. The game begins with a letter B.
when guessed please continue with this forum topic by adding more game storylines/beginnings to be guessed.

From a quick Googling, is it Bayonetta?

Hm, here's mine. I made a quick chiptune arrangement of part of the soundtrack, see if you can figure it out from this. Google won't help you this time.

http://bit.ly/11o9uIU

Somehow I already knew what this was going to be before I even downloaded it. It's Bynn the Breaker, from Bastion (although to be honest the filename didn't give it away).

I can't really think of a good one, but I'll see if this hint is good enough: in this game, you need to save a princess, but it's not Mario.

Puggsoy: Braid

Speaking of time-related games, in this game, you travel back and forth through time to save your planet from being devoured by an evil monster.

I think we need another hint, MOM.

Mmkay then. It's an SNES JRPG.

Maybe another?

Made by Square Enix? Multiple endings? I'm kind of just googling keywords that I've said so far and spitting back information I'm finding on the game here at this point...

Now I think I have it...Chrono Trigger.

But Google results aren't a good indication, they're shaped by what you've searched for before. Typing in "World" on my machine has "World of Goo" -> 2DBoy's site as the top prediction.

[Clues pending, stuff needs to upload...]

Yep, Chrono Trigger it is.
As far as that goes, I think that's just for suggestions for completion. The first hit for "World of Goo" should be the same for everybody.

Okay, my videos are glitching out, so here's a different recording while I try to fix the others. Sorry, this one isn't my best, I shot it quickly while my piano was being repaired because it looked cool with the casing off.

Name the game that this music comes from. The game is pretty famous, but this isn't its best-known song.

I answered on the Youtube comments. Don't scroll down unless you give up.

"Mmmm Mmmmm MMMMM weeeEEEE!"

That sounds like a quote from Little Inferno?

Oof, can't say I remember that song from anywhere, yet it sounds strangely familiar. I'm gonna take a guess and say Vessel, since it sounds like it would fit in there.

Close, but not quite. Red was correct, it's Geothermal from Cave Story (aka the music that plays in the core room before you find the robot, and then again after you defeat the core if you didn't find the Tow Rope). I really like that piece, but it isn't really written for piano.

Oh OK, I'm not even that far yet. I always play for a bit, and then there's a boss I can't beat so I take a break. Is it just me or does Cave Story have unusually frequent boss battles?

Anyway yeah Red can do the next game then.

Ah, now I need to do another video! It should be up by tomorrow...