When somebody is calling you, the network send the message to all Base Station, and your phone answer "I am here". The network assign a channel between your phone and the Switch, and complete the channel with the calling phone.
When you start moving, the phone measures the power and check which Base Station is near to change. The network orders the Handover from one Base station to the other.
The Cell Phone "talks to" Base Transceiver Station (BTS); the BTS is linked to Base Station Controller (BSC) and the BSC is linked to MSC or Switch. Switches are interconnected to other National, Cellular and International Switches.
(A Switch is a Phone switching central).
THe interconnection between BTS-BSC, BSC-MSC and MSC-MSC is by Fibre Optics, Microwaves, Satellites, twisted cupper pairs...
Hope it helps.