Depletion regions are regions of doped semiconductor which have been depleted* of mobile charges (electrons for n-type semiconductor or holes for p-type semiconductor). Since the electrons or holes have left the depletion region, the region is electrically charged. p-type depletion regions are negatively charged (due to acceptor ions) and n-type depletion regions are positively charged (due to donor ions).
* In reality, free charges exist in the depletion region, but at many orders of magnitude lower concentration than in the electrically-neutral regions.
Depletion regions exist in many semiconductor devices. The first depletion region many students encounter is that in the p-n junction (see writeup there). A metal/semiconductor junction (ohmic contact or Schottky diode) has a depletion region very similar to that of a p-n junction; the only difference is that the depletion region exists only on one side (the semiconductor side) of the junction. The MOSFET has a depletion region under the gate oxide that is formed before the MOSFET becomes conducting (turns on).
The fact that mobile carriers leave a region, forming a depletion region, is due either to diffusion or electric field. In the p-n junction and metal/semiconductor junction, the depletion region exists because of hole and electron diffusion across the junction. Eventually the electric field created in the depletion region inhibits any more diffusion, and equilibrium is reached. In the MOSFET, a voltage on the gate acts as a force on carriers beneath the gate, pushing them away and creating a depletion region.
The depletion region behaves as a capacitor since the lack of mobile charges makes it essentially a dielectric (insulator). The width of a depletion region is the following:
W = (2εV/qN)1/2,
where ε is the dielectric constant of the semiconductor, V is the sum of internal and applied voltages, q is the electron charge, and N is the doping concentration of the semiconductor. In the case of the p-n junction, 1/N must be replaced by (1/Na + 1/Nd), where Na is the acceptor concentration on the p-side and Nd is the donor concentration on the n-side. Notice that if one side is more heavily-doped than the other, only the lightly-doped side contributes to the width of the depletion region. Such a junction becomes more like a metal/semiconductor junction (one-sided), since a heavily-doped semiconductor has properties similar to a metal.