A PBX is a small version of the phone company’s central switching office. All PBX’s are automatic; no operator is needed. A private telephone network is used within an enterprise. Users of the PBX share a certain number of outside lines for making telephone calls external to the PBX.
Broken down, a PBX is;
Private in that the end user owns it.
Branch meaning a small phone company’s central office.
Exchange means the phone company’s central office