Ok, I'll take a stab at this,
Part 135 flying has two main divisions:
First is nonsched, commonly known as charter or "on demand". these operations don't have a published schedule. They fly either passengers or freight in aircraft with less than 9 passenger seats and I think a payload capacity of 7500 pounds or less. Operationally speaking all that is required is a properly certified airplane, properly certified pilot and a customer. This is overly simplified, my goal is just to convey the basics. Also always remember with the FAA where there is a will there is a waiver. I know of at least one company operating 19 passenger seats on a part 135 certificate.
Scheduled 135 operations have a published schedule, and a slightly more complex operations structure. The operation is still limited to 9 passenger seats and a payload capacity of 7500lbs or less.
Part 121 has a whole different set of regulatory requirements. There are also two divisions.
Scheduled part 121 flying is passenger or freight in bigger airplanes, with more than 9 passenger seats or more than 7500lb payload. There are heightened training requirements, airport facility requirements, operational structure is also strongly regulated. There is a requirement for a Dispatcher who releases each flight, provides information to the flight crew, trouble shoots operational problems, and most importantly maintains operational control over each flight. This is done by a series of checks and balances all centering around the dispatch release. Don’t get me wrong here, the captain is always responsible for operating the aircraft, but must confer with the dispatcher when it comes to dealing with changing weather conditions, operational issues with the aircraft. i.e. MX
but I digress slightly.
Nonshed 121, also referred to as supplemental operations are basically charter ops with aircraft seating more than 9 pax or more than 7500lbs of payload. I'm noticing a continuing theme here... With supplemental ops, the director of operations takes on the operational control function eliminating the requirement for a dispatcher. The DO has the ability to delegate this function, and I have not yet run into a supplemental operation that does not incorporate a dispatcher.
Now for the waiver of liability: It’s after 3 am here and I'm only a pilot, furloughed at that. So feel free to correct, modify, or completely disregard what I have said.