Technically, an independent (indie) studio is simply a development studio that is not owned by a game publisher. But at the heart of the matter is really artistic freedom, or lack there of with big commercial titles.
More of less, the technical definition can really get you nowhere. Blizzard publishes their own games, simply because they are horrendously rich; And Minecraft's Notch is seriously thinking about/already decided to become a publisher and is also quit well to do after the success of Minecraft. But the answer to both questions of "is this company indie" is obvious to anyone; Blizzard is not and never can be, and Notch is and always will be. And then there is the issue of subsidiarity companies; Publishers that do not own the "indie" development team but completely or partially bankroll the development of a project; "indie" development teams that develop a project completely internally but go the traditional route of a publisher; And even development studios that are still run traditionally, with the traditional middle men, advertisers, and accountants, put in charge of the actual developers.
So now that I have shown that the official definition is far from satisfactory let me try to give you one that is. Indie is about artistic freedom and a rejection of conventional practices, values, and style. Indie development is about making the game you want to make and not one that will appeal to the maximum amount of people possible. It is about the coders, designers, writers, and drawers all having a stake in the company over and above their next pay check and all having a voice in the game design. It is about small teams of friends, and single man operations. it is about games that could not be commercialness viable, free games, pay-what-you-want games, and a rejection of profit. It is about games as art, with visuals just as appealing as paintings or sculptures. And while you don't need all of these things going for you at the same time to be indie, it does help; Participially the profits part. While it is a big part of the indie scene to reject the conventional greed of mainstream developers no one would ever say that no one can or should make a profit as an independent developer, but it does make them different and less indie then developers that give away their games. Not that anyone should think less of them for it, they are less indie (I think that is irrefutable), but they provide a great middle ground and often produce great games that could not be produced by either non-for-profit indie developers nor mainstream studios.