jobydon,
First, ignore cb_boarder04. He's wrong, it would be Part 121, not part 125, and it has nothing to do with whether the operations are scheduled or not. There are a great number of Part 121 airlines which operate exclusively non-scheduled flights.
The key to which part the operations falls under is whether or not it is "common carriage" . First, you have to understand what "common carriage" is. In a nutshell, common carriage means that you will carry persons or property for hire for the general public. In other words, you will sell tickets to any persons or organizations who wish to buy your services, or your will carry cargo for anyone who has a plane load of cargo and is willing to pay your rates. That's common carriage, you do business with whoever has business.
If you think about it, the vast majority of people offering air transportation are engaged in "common carriage". There are relatively few operators engaged in air commerce who are not engaged in common carriage.
There are a few ways in which you can be engages in other than common carriage:
The first is if you are not offering services to the general public, but only to one or a very few limited customers. Suppose that you had an exclusive contract to haul cargo for a mining company to 3 of their remote mines. That was all the flying your did, you performed the flying on a long term contract, and you did not do any flying for any other customers.
That would be an example of an operation that did not involve common carriage.
If someone else showed up in your parking lot with a truckload of cargo and a bag full of cash and requested that you fly his cargo to a destination of his choosing, you would tell him no (or, if you did agree, you would then be engaging in common carriage)
Another example of operations not involving common carriage would be if the operator owned the property being carried. My previous employer was engaged in this business (and common carriage, as well) they had two operating certificates, one Part 121 and one Part 125 and two fleets of airplanes. The Part 125 certificate and airplanes were used for delivering bulk fuel to remote locations in Alaska. While the fuel business had many customers, it was not common carriage, because when the fuel was being carried in the airplanes, it was owned by the company. We had our own bulk fuel business and the point of sale to the customer was at the airport of destination. So, it wasn't common carriage, because we were not carrying the property of *others*. The part 121 certificate was used for scheduled and non scheduled cargo flights, which were common carriage.
The question of who the fuel belonged to was not a semantic point or an end run around the regulations. Flying your own property in your own plane does not constitute common carriage. We did have one contract to carry bulk fuel which involved fuel that involved fuel which was owned by the customer. That particular operation *was* common carriage, and we could not fly it under Part 125. We installed bulk tanks in one of our Part 121 planes and those flights were done under Part 121.
Anyway, the bottom line is that the difference between Part 125 and 121 has nothing to do with schedules, it has to do with whether or not the operation is common carriage.