Actually, an independent film is defined as "a film financed in part by pre-sales to distributors, often on the basis of a script or a star attached to the project". The pre-production phase is done independently, without finance (ie. writing the script, casting, recruiting the creative crew etc.) The completion of the project is risked by a few financiers who believe in the project's success based on the creative materials provided. This differentiates from the studio system where a team of creatives are employed to create films within studio control. Critics and fans tend to look down on this system because creative control is dictated by producers that make business decisions in the creative process.
Below is a great article that discusses how the Lord of the Rings Trilogy was funded independently. It's bizarre and inspiring to think that such a large franchise was funded in this way. I believe the success of this series was Peter Jackson's ability to retain creative control which was possible by fund-raising on the stipulation that his ideas remain intact. However, it's difficult get funding in this way without a solid reputation for being a successful filmmaker. Read and let us know what you think.