Anything (band with independent label, style, person, etc.) that is 'independent'.
Bands are considered indie if they are independent label and make almost no money at all.
Kids generally call themselves indie when they shop at thrift stores. After their escapades in the thrift store, they then buy thousand-dollar Nikon cameras and a polaroid. They take loads of pictures from strange angles, take pointless pictures (such as shoes), and take provocative pictures (but to them, it's not provocative - it's indie).
These people then bitch to normal people about how 'normal' is overrated and that listening to mainstream music shows that you have no individuality.
Apparently, the only way to be an individual is to buy cheap-ass clothes that have been worn by who knows how many people, waste your time and money, and preach to others.
Indie Guy: What are you listening to?
John: Dynamite by Taio Cruz - man this shit has a sick beat.
Indie Guy: Wow. Way to completely turn over your individual soul and personality to the corporate music industry. Find mental maturity through real music by bands that nobody knows.
John: Dude. Shut up.
Indie Guy: I'm gonna go take some pictures of some stupid shit like dirt now.
John: Stupid ass indie kid.