What is a power divider in a truck when and how it should be used?

On a tandem truck, you have two configurations - single screw, where the driveshaft goes directly from the transmission output directly to the differential of a live axle... the other axle will be a dead axle, and does not propel the vehicle.

The other configuration is known as twin screw, and this is what a 6x4 truck is. Both axles are live axles, and the driveshaft goes from the transmission output to the power divider input. The power divider is essentially a differential, which ensures that power is distributed evenly to both output shafts (one going to each axle differential).

It's always in use. As for the lock, you use it as little as possible. Icy conditions, off-road use, or times when an axle is lifted off the ground would typically be when you'd use the PDL.