A car's engine generates power, which is then passed through the transmission and sent to the wheels. Before it gets there, one more system has to step in to make the wheels spin and get the car moving. This system is called the differential, and it is in charge of three different things.
The differential takes the power passing from the transmission and directs it to the wheels. It also provides one more gear reduction to slow the transmission's rotational speed enough so the wheels don't spin faster than they need to. Perhaps most importantly, the differential is responsible for adjusting the axle rotation rate so the wheels can operate at different speeds - which is how the system got its name.
When a car turns, each wheel follows a unique path. The wheels of a driven axle are connected so the engine and transmission can control them simultaneously. Differentials allow them to turn independently so one wheel does not slip. On cars with only one driven axle, there is only one differential, but all-wheel-drive vehicles need one at each axle.
Front-engine rear-wheel drive cars have a drive shaft to connect the transmission and the differential. The setup for all-wheel-drive vehicles is similar, but there is also a center differential located on the drive shaft. If you have a car that can be switched into four-wheel drive mode, it will not have a center differential, which makes turning on pavement a challenge.
In an open differential, the most basic setup, a large ring gear is attached to the driven axle. Two pinion gears connected to the ring gear are interlocked with gears on the axle shaft. The whole system is set in motion by an input pinion, which is spun by the transmission. The input pinion is smaller than the ring gear, which provides the final gear reduction.
When the vehicle is traveling in a straight line, the pinion gears spin with the axle, and both wheels turn at the same speed. As soon as you turn the steering wheel, the pinion gears begin to spin independently. This causes the different sides of the axle to rotate at different speeds and allows the wheels to operate on their own.