There is an electronic solenoid controlled by the ECU which controls the Boost level. Remapping the ECU alters the boost level by changing when this boost solenoid operates.
It basically connects the Positive pressure side of the turbo back to the wastegate. When the pressure on the positive pressure side reaches a certain point the ECU operates the solenoid allowing the wastegate to open.
The wastegate valve is mechanical, but the control of boost pressure going to the mechanical valve is electronic. In the early days of turbocharging it would have been a purely mechanical system.
Thats a basic explanation anyway.
There is a good article on Turbos on Wikipedia.
HTH Simon.