'Indie' originally meant any band signed to an independent rather than major label. But there is a "sound" that those kind of bands generally have, and even when they aren't signed to an independent label, they still sound raw and "self made". So the term "indie" has kind of passed itself onto describing the music, rather than the music label.
There are different sub genres of indie, and they all have different characteristics. For example Indie can range from Indie rock to Indie pop. Indie pop normally including more acoustic characteristics.But I would say that the one thing all indie music has in common is this: authentic rather than manufactured, sounding, and it should have obscure or different lyrics rather than typical ones you hear from big, arena bands.
Grunge bands like Nirvana were a form of Indie Rock. That's another thing, indie tends to be more mellow and emotional than rock music that edges towards metal. Brit rock, like Oasis and Blur, are indie too. Modern bands like MGMT and Vampire Weekend are very "different" sounding, they mish mash lots of genres together.
I feel indie is almost controversial and contradicting, though I do really like it, because everyone doing Indie are supposed to be "Independent" yet they all follow the same characteristics.