The specific definitions of hosted PBX can differ from one provider to another. Specialized providers have added their own unique twists and definitions to the basic, technical meaning of hosted PBX as a means of branding and marketing. The bottom-line doesn't change, however, that hosted PBX is simply that: PBX that is hosted.
The telephone switchboard hardware known as Private Branch eXchange (PBX) is usually located in the premises of the user. The difference with hosted PBX is that this piece of hardware is located or "hosted" elsewhere, typically along with the call platform. The consumers then simply typically connect to the host via IP.