Plates move because they are being rafted by the mantle underneath. The mantle underneath is convecting, causing the surface to move (and thus plates to move), and the reason why the mantle is doing this is to release the heat from the interior of the Earth. However, since plates are rigid (they're made of rock, after all), they tend to catch on each-other, and don't move until there is enough stress to break them and cause them to shift catastrophically - an earthquake.

Earthquakes produce several different types of waves, akin to sound waves, that travel from the point of origin of the quake. These different waves have different strengths and different speeds. The fastest are compression waves, followed by shifting(back-and-forth) waves, and finally surface rolling waves.