Indie is a developer that's independent and typically small scale.
The appeal of this kind of developer is simple: No game starts out bad, or kicks off with a shitload of bad ideas. Those things get implemented as deadlines get closer, production costs rise, and people lose sight of their first projected product. Sometimes it's for the better, and oftentimes it's for the worst.
Indie games remove that by keeping production costs low and staying independent. Obviously this means you can't have the triple-A quality control in graphics, coding, and technology. But they serve as strong proofs of concept and can be a cheap alternative to the sea of generic shooters everyone likes complaining about. Furthermore, with less money invested by less people, developers are free to take more risks, which is why there are so many "artsy" indie games in the market.
Of course, there are the bad kind of indie, which Nintendo has affectionately dubbed, "garage developers". Developers like those guys behind Angry Birds or The Wonderful End of the World contribute nothing to the industry because they're essentially just ripping off of already-successful games or game genres and selling it to you at a cheap price. This is a worthless venture that contributes nothing to the industry besides a boring, stable business venture. The ever-popular iPhone game industry is full of these guys.