Generally music with a more obscure sound. Indie music consists of musicians who aren't in it for the big label, but to make good music, or music they like. Genres include folk, rap, electronica, acoustic, etc.
Anybody who considers Modest Mouse to be an indie band is incredibly ignorant.
The genre should not be mistaken with "indie kids". (Who the hell labels themself as an indie kid, huh?)
Indie music doesn't ONLY consist of acoustic guitar, whiny voices, and shouldn't be mistaken with teenagers who wear black rimmed glasses, put the Juno album on repeat, wear converse, and hang out at Starbucks.
Try some Gregory and the Hawk, he Dreams of Sleep, Robbie Snell, Wesley Bunch.