This is a system generally used for distributing power to the wheels of a vehical. In a vehical with paired wheels, it can be demonstrated that when cornering, the wheels on the outside of the corner will travel furthar than those on the inside. Because of this, they need to turn faster.
Now if you had a simple axle accross the two wheels, if you turned a relatively heavy vehical at speed - the torsion caused by the difference in speed between the two wheels would cause something to snap - bad news.
What a differential will do is allow those wheels to actually turn at different speeds, and if the output to one wheel is limited - then more power will be output on the other wheel. This is why if you get one wheel stuck in mud, the other wheel spins at twice the speed.
This is done by houseing 3 or four small bevel gears in a fully rotating housing. The housing itself has a toothed outside, and the power is input via this. However - the houseing can also rotate freely round its axis. Along its axis are two axles, one at each side, each connected to a bevel gear. The third (and fourth) bevel gears are then attached to the housing perpendicular to the output gears. When the axle is rotated - the latter two gears rotate, and if there is no difference in the wheels - the bevel gears will simply turn with the housing. However - if one output shaft is stopped - then the latter two bevels will have to turn on their axes to compensate, this is then transmitted to the the opposite output gear and shaft - which will rotate at twice the speed.
An animated gif diagram will be added in due course.
To be continued..