PBX stands for Private Branch Exchange, which is a private telephone network used within a company or organization. The users of the PBX phone system can communicate within their company or organization and the outside world, using different communication channels like Voice over IP, ISDN or analog. A PBX also allows you to have more phones than physical phone lines (PTSN) and allows free calls between users. It also provides features like transfers, voicemail, call recording, interactive voice menus (IVRs) and ACD call queues.
PBX phone systems are available as Hosted or Virtual solutions (sometimes also called Centrex), and as in-house solutions to be used on your own hardware.
PBX phone systems are usually much more flexible than proprietary systems, as they are using open standards and interfaces.