The Commercial Pilot certificate, issued under 14 CFR 61.133, allows you to act as pilot in command of an aircraft carrying persons or property for compensation or hire, and to be paid to fly. This is the legal threshold between a hobby and a profession.
However, the certificate alone does not authorize you to operate a commercial business carrying the public. Most revenue passenger or cargo operations require the operator to hold an air carrier or operating certificate under Part 119/121/135. As a commercial pilot you supply the piloting privilege; the operation is governed separately.
Think of it this way: the certificate makes you legal to be compensated; Part 119/135 makes the operation legal.