Short answer from FAR 298
Commuter air carrier means an air carrier as established by §
298.3(b) that carries passengers on at least five round trips per week on at least one route between two or more points according to its
published flight schedules that specify the times, days of the week,
and places between which those flights are performed.
(b) There is hereby established a classification of air carriers,
designated as “commuter air carriers,” which directly engage in the
air transportation of persons, property or mail, and which:
(1) Do not directly or indirectly utilize large aircraft in air
(2) Do not hold a certificate of public convenience and necessity;
(3) Carry passengers on at least five round trips per week on at least
one route between two or more points according to its published flight
schedules that specify the times, days of the week, and places between
which those flights are performed;
(4) Have and maintain in effect liability insurance coverage in
compliance with the requirements set forth in part 205 of this chapter
and have and maintain a current certificate of insurance evidencing
such coverage on file with the Department;
(5) Have and maintain in effect and on file with the Department a
signed counterpart of Agreement 18900 (OST Form 4523) and comply with
all other requirements of part 203 of this chapter; and
(6) Hold a Commuter Air Carrier Authorization issued in accordance
with subpart E of this part.
Almost all public carrier operations are under part 119, which specifies certification requirements with reference to 121, 135, etc. Public, meaning they have many customers as opposed to a negotiated contract for special services.
Part 135 operations are essentially small in nature, there are several sub-categories of 135 operations from an owner-operator up to large fleets of small airplanes. The exact restrictions will depend on the sort of flights, being on-demand or regularly scheduled(5 or more per week). A 135 with a part 298 authorization is an air-taxi and may have more than 5 trips per week per city-pair.
Large-aircraft operations will be 121(scheduled public direct operations) or 125(private. eg a corporate 737 moving employees)
Part 380 operations are non-scheduled(by definition, though they likely have a set departure date and time.) "public charter flights". 380 is intended for the indirect carrier, also called a charter organizer which acts as a third party between passengers and operators of aircraft on a whole-flight basis. This is a company that arranges charter flights and sells tickets on those flights but does not generally directly operate the aircraft. Although both halves, indirect organizer and direct operator, may in some cases be owned by the same company they are required to be financially separated units. Passengers of 380 flights must sign a notice explicitly indicating that this is not a normal scheduled carrier operation(121/135). Regularly flown part 380 flights of part 135 size aircraft may need a part 298 authorized direct operator.
Administratively a 135 has a few less required company officers(legally responsible parties) and simplified expectations for their operation specifications. As compared to 121.
See FAA Advisory Circular AC_120-49A for more guidance on the certificate application and qualification requirements of 121 and 135.
From an FAA letter of legal interpretation:
Section 380.2 defines a "U.S. public charter operator" as an indirect
air carrier that is authorized to engage in the formation of groups
for transportation on public charters. Section 380.2 also defines a
"public charter" as a one-way or round-trip charter flight to be
performed by one or more direct air carriers that is arranged and
sponsored by a charter operator. Also, §380.2 states "a charter flight
is a flight operated under the terms of a charter contract between a
direct air carrier and its customers. It does not include scheduled
air transportation, scheduled foreign air transportation, or
nonscheduled cargo air transportation." All public charter passengers
must sign an operator participant contract (§380.32) acknowledging
that they are being transported on a charter as opposed to a scheduled
flight. So, although the operations may look like scheduled
operations, the passengers are treated as charter passengers in an