Architecture of the GSM network
A GSM network is composed of several functional entities, whose functions and interfaces are specified. Figure 1 shows the layout of a generic GSM network. The GSM network can be divided into three broad parts. The Mobile Station is carried by the subscriber. The Base Station Subsystem controls the radio link with the Mobile Station. The Network Subsystem, the main part of which is the Mobile services Switching Center (MSC), performs the switching of calls between the mobile users, and between mobile and fixed network users. The MSC also handles the mobility management operations. Not shown is the Operations and Maintenance Center, which oversees the proper operation and setup of the network. The Mobile Station and the Base Station Subsystem communicate across the Um interface, also known as the air interface or radio link. The Base Station Subsystem communicates with the Mobile services Switching Center across the A interface.
As John states, he presents a generic GSM architecture. Lucent, Ericsson, Nokia, and others feature their own vision in their own diagrams. But they all share the same main elements and parts from different vendors should all work together. The links below show how these vendors picture the GSM architecture. You can remember the different terms much better by looking at all these diagrams.
Lucent GSM architecture/ Ericsson GSM architecture / Nokia GSM architecture / Siemen's GSM architecture

Figure 1. General architecture of a GSM network
SIM: Subscriber identify module.
ME: Mobile equipment.
BTS: Base transceiver station.
BSC: Base station controller.
HLR: Home location register.
VLR: Visitor location register.
MSC: Mobile services switching center.
EIR: Equipment identity register.
AuC: Authentication Center.
UM: Represents the radio link.
Abis: Represents the interface between the base stations and base station controllers.
"A": The interface between the base station subsystem and the network subsystem.
PSTN and PSPDN: Public switched telephone network and packet switched public data network.