Common carriage is the key concept that separates private from commercial operations. The four elements are: (1) holding out to the public, (2) to transport persons or property, (3) from place to place, (4) for compensation. An operator that holds out to carry the public is a common carrier and needs a Part 119/135 certificate.
Holding out means signaling a willingness to serve anyone — advertising, word of mouth, a reputation for providing transportation. Certain operations are explicitly excepted or governed by Part 91 subpart F / 91.501 (e.g., some operations of large/turbojet aircraft, time-sharing, interchange). The classic Part 91 commercial jobs that do not require 135 include aerial photography, banner towing, crop dusting (Part 137), pipeline patrol, flight instruction, and ferrying — because they are not transporting the public from place to place.