IP PBX is short for "Internet Protocol Private Branch Exchange". A PBX splits one main office phone number into several extensions. For instance, when someone calls an office and is told “Press 1 for John, press 2 for Monica”, that caller is interacting with the PBX, and that PBX is going to connect his or her call to John or Monica.
An IP PBX simply uses the Internet to connect these calls instead of traditional (and expensive) telephone wires. An office would need some kind of PBX or SIP solution in order to implement cheap VoIP phone service in an office. An IP PBX is located and maintained in-house.