Differential lock; diff-lock; locking differential; locker.

What does it do?
Keeps the axle "locked" so that both wheels on the axle rotate at the same speed. 
You can get front, rear and central differential locks.

Why do you want to have one?
To keep power going to tires which have traction.  Good for using in tricky situations on soft uneven surfaces (mud, snow, sand).

Example:
If you have one wheel up in the air and the other tires are bogged down, without a locked differential the tire which has no traction will spin freely, taking power from the other wheels.
With a locked differential the free wheel will still turn, but only at the same speed as the other tire(s) it is locked to, thereby reducing loss of power.

Why don't you want to have one?
An open differential will allow wheels to turn at different speeds, which helps reduce tire scuffing when going around corners.  If you leave your differential locked on surfaces with good traction (like tarmac) you will cause torque to build up in the transmission so that components wear and / or snap.  Make sure to dis-engage the diff lock when you reach a hard surface. 

Still don't get it?
http://www.youtube.com/watch?v=K4JhruinbWc

http://www.youtube.com/watch?v=RcRBYpccP6g