In the past twenty years or so, major studios have set up "boutique distributors" for independently financed films with usually a more specialized intellectual focus. These included Miramax (Disney, separated from The Weinstein Company/Dimension), PictureHouse (New Line, and this used to be Fine Line), Warner Independent Pictures, Sony Pictures Classics, Paramount Classics (now Paramount Vantage), and Fox Searchlight Pictures.
A number of really independent companies cropped up, including ThinkFilm (Canadian), Magnolia, LionsGate (which absorbed Artisan and also has an interest in Lifetime) and Strand (which has a lot of GLBT material, as does TLA).
The business relationships among these companies are complicated. Often a film has one distributor in the US, another internationally, and still another for DVDs. Some companies (like Miramax) double up as production companies and distributors.
Disney uses Buena Vista for most of its domestic distribution, and has Touchstone, Hollywood, and Walt Disney Pictures itself as "brands". Sony does something similar. Columbia, with the Statue of Liberty trademark, is the central brand, followed by MGM, which it absorbed (but it kept the Lion), TriStar, Screen Gems (for horror and action). United Artists, which Tom Cruise is taking over as CEO, was long a "mainstream" brand for large budget independent films, but MGM made it into the "boutique label" and apparently it will maintain an independent focus under Sony/Cruise. Paramount absorbed Dreamworks, and sometimes uses the Dreamworks company for large films with a more intellectual focus (like Clint Eastwood's "Flags of our Fathers" or "The Last Kiss" with Zach Braff -- anything with Zach is automatically independent!). Paramount released Alejandro Gonzalez Inarritu's "Babel" with both it's mountaintop Paramount trademark and Paramount Vantage.
Recently, independent art houses like Landmark Theaters have been showing more films marked under major studio brand labels, like "Little Children" from New Line, and "Stranger than Fiction" from Columbia, as well as "Running With Scissors" from TriStar.
And there are some production studios like Participant and Liberty that sometimes distribute their own films.
So, what is an independent film? YouTube?