Part 135 is for air taxi, a.k.a. charter flights. Example: A charter company with a couple of lears and a G-550 that seek customers on a per-hour basis to charter the whole aircraft. This isn't the only example, but is the most common type of 135 carrier.
Part 121 is for airlines. Example: American Airlines, Delta, Comair, ExpressJet, etc.
Part 91 is for general aviation flying, i.e. non-commercial.
There are others, but those are the 3 main chapters regulating flight operations.
Tim