I would like to devote this thread to listing and discussing everyone's favourite anime character/image songs. I did contemplate posting this to /music/, but they'd probably just tell me to gb2here, so I'll skip that step.
My current favourite is "Shiawase Negau Kanata Kara"/幸せ願う彼方から from Lucky Star, sung by Shimamoto Sumi. I don't mind admitting that it almost never fails to make me teary-eyed, though that might be because of its association with a very well-done MAD I saw on Nicovideo some months ago. MAD is here, no idea if it's on YouTube yet: http://www.nicovideo.jp/watch/sm1594419
That MAD is so oldschool, very nostalgic. Yeah, that character song almost never fails to get me. Currently, however, my favorite character song is probably Kaname Madoka's "Mata Ashita" (since it's still relatively new, most uploads are deleted on video sharing sites I think). Like any good character song it's directly related to the series, and the lyrics are definitely enough to get me worked up.
For something a little bit more light-hearted (and older), it's Takatsuki Yayoi's Kiramekirari. It's from idolm@ster, so might be cheating, but still a character song nonetheless!
http://www.nicovideo.jp/watch/sm841196
I think Rika Furude/Yukari Tamura's song "Mugen Kairou" from Higurashi no Naku Koro ni is just about the best character song out there. Great music, more than a notch above the typical uninspired composition, and fantastic lyrics too. It's a particular contrast after hearing at the character songs for the series, which are all dumb jokes...
As a guilty pleasure, though, I like Akari Himuro/Ema Kogure's "Destiny" from Night Wizard. Full of Engrish, objectively not that great a song, but something about the chorus just gets to me. It feels so... heartfelt. Ema Kogure actually created and played Akari's character in the tabletop RPG sessions the anime was based on, so maybe it is more so?
http://www.youtube.com/watch?v=rSvsaR4RSTA
Nii nii suki.
I also want to add Alice Carroll from Aria the Origination Ep-09. She sang her canzone "Lumis Eterne". Ryo Hirohashi did really well for Alice Carroll's singing and it's really great as well as the anime itself.
http://www.youtube.com/watch?v=TGmAqjiA9ko&feature=related
If Alice Carroll has made a great performance with her "canzone" you can also listen to her senior Athena who was voiced by Tomoko Kawakami (RIP), err but the singing voice was Kawai Eri (RIP).
http://www.youtube.com/watch?v=fSs6pdCSkGQ