indie, alternative, and rock
First band that came to mind when I read this was
Deas Vail, which just so happens to be one of my favorite alternative indie rock bands ever. Absolutely beautiful music. I also agree with with anyone who said The Mars Volta; they are crazy amazing and even if you aren't too sure about them at first, they will grow on you. Other than most of the bands that have already been mentioned, I also suggest:
Sparta
The Alkaline Trio
Andy Mckee (If you've ever even touched a guitar at any point in your life, you will appreciate this guy. Check him out
http://youtube.com/watch?v=Ddn4MGaS3N4)
Billy Talent
Circa Survive (I find that they sound very similar to Coheed and Cambria.... Might just be the high pitched vocals)
Hundred Reasons
Rush
Social code