How does a limited slip differential work?

Its usually a pair of clutches (like a clutch pedal in a manual car). The more torque present on the driveshaft the harder the clutches are held together. This allows for some independent slip within the drive wheels. In modern cars the control of the clutch is electronic rather than mechanical, such as Mitsubishi's Anti-Yaw-Control (AYC).

See: http://en.wikipedia.org/wiki/Limited_slip_differential
http://en.wikipedia.org/wiki/Active_Yaw_Control