I'm going to post some of what I consider indie. I've seen some already mentioned so I won't post those.
Here's some good music I've been listening to lately, not hard at all really but somebody here will most likely enjoy them. Everything I link to I highly recommend
This is a really sad song by a fairly well-known band called Eels, the whole album is amazing. Eels is an alternative/acoustic indie rock band.
http://www.youtube.com/watch?v=z3S3QSNOE88
This next song is by one member of the band Animal Collective, an experimental group. This whole album is also highly recommended (album is called Down There).
http://www.youtube.com/watch?v=Zz2VGEQ4x-Y
This next song is by Strand of Oaks, really beautiful album. Genre is folk I believe. This whole album is also absolutely amazing.
http://www.youtube.com/watch?v=qQzDW1IdgQ8
This next song is by a gem of a band called Snowmine, I can't understand how they're not more known already. Genre is indie/psychedelic, or at least that's what I've been told. Album is called Laminate Pet Animal, the whole thing is amazing, free download on their bandcamp page. http://snowmine.bandcamp.com/
http://www.youtube.com/watch?v=iIqqcjMzavM
This next album is also pretty well known. By an alternative/post-punk (or something) band called The Replacements.
http://www.youtube.com/watch?v=RJcCzWcgPsY
Here's my post from the music recommendations thread, only things not "indie" on there are The Replacements and Avey Tare but you might also enjoy those, so you should check them out as well.
I'll post some more as well:
http://www.youtube.com/watch?v=ohgr51CqliY
http://www.youtube.com/watch?v=KMMjEOY3VGk
http://www.youtube.com/watch?v=QXYM6-X8c3o
http://www.youtube.com/watch?v=oPJw9-uWIRE
http://www.youtube.com/watch?v=8YCLBL4LEkc
EDIT: If I'm gonna post Titus Andronicus, might as well post AJJ as well.
http://www.youtube.com/watch?v=iDH_4lalRtQ
EDIT 2: If I'm gonna post AJJ might as well post WDU.
http://www.youtube.com/watch?v=AagXbraxPK0
http://www.youtube.com/watch?v=o8YHVQGfFLs
http://www.youtube.com/watch?v=f0O7oVK2xi0