Mobile network is the term we give to distributed radio systems designed so that many users who may or may not be moving around can share a radio band and communicate amongst themselves and to the PSTN.
Such network service providers construct networks by purchasing network components from vendors and deploying and integrating the components. This includes the physical antennas on towers or on top of buildings, racks of radios in the Base Station Transceiver (BST) at the base of the antenna, and the Base Station Controller (BSC) which interfaces the base stations to the mobile telephone switch.
The mobile communication network service provider's switch is connected to the Public Switched Telephone Network (PSTN), so that you can call someone on a landline from a cell phone.
The Home Location Register is a database which keeps track of who is on the air and where they are (or last were) via radio control channels and Electronic Serial Numbers (ESNs).