The primary role of the differential is to transmit the engine power to the wheels. Inner wheels need to turn slower than the outer set so the differential splits the torque enabling them to rotate at different speeds. All modern vehicles have them but the location of the differential is dependant on whether you driver a front or rear wheel drive vehicle.
The most common sign the differential is experiencing problems is sound. A grinding or growling noise which can become louder as you accelerate or turn a corner common causing friction.
Before you rush out to replace the differentail, it is worth checking the fluid levels to ensure you do not have a leak.