What is the difference between an air carrier and a commercial operator?
From 14 CFR 1:

Air carrier means a person who undertakes directly by lease, or other arrangement, to engage in air transportation.



Commercial operator means a person who, for compensation or hire, engages in the carriage by aircraft in air commerce of persons or property, other than as an air carrier or foreign air carrier or under the authority of Part 375 of this title. Where it is doubtful that an operation is for “compensation or hire”, the test applied is whether the carriage by air is merely incidental to the person's other business or is, in itself, a major enterprise for profit.

To make this more clear FAA Order 8900.1, Volume 2, Chapter 2, section 2 says:

2-128    TYPES OF CERTIFICATES UNDER PART 119. There are two, basic types of air operator certificates (AOC) issued to U.S. applicants who will conduct operations in common carriage. See Table 2-4, Certification for a summary of certificate requirements.
A.    An Air Carrier Certificate. This certificate is issued to applicants that plan to conduct interstate, foreign, or overseas transportation, or to carry mail.
B.    An Operating Certificate. This certificate is issued to applicants that plan to conduct intrastate transportation.

From your perspective it isn't a too important distinction.  As a commercial pilot all that matters to you is that both air carriers and commercial operators need to have operating certificates.  As a commercial pilot you can be hired by someone who has an operating certificate but you cannot provide the same services on your own unless you yourself have an operating certificate (e.g. a charter operator can hire you, but you cannot operate a charter service).
As a commercial pilot you can be paid to act as PIC and can engage in the commercial operations that are excepted from operating certificates in 14 CFR 119.1(e).  You cannot act as either an air carrier or a commercial operator without complying with 14 CFR 119, which requires that you have an operating certificate.