Heat
Heat plays a major part in the movement of tectonic plates. Deep under the surface of the Earth, the closer to the core you get the hotter the ground. This heat makes certain layers of rock soft and malleable, which at the extreme appears as magma, but even away from volcanoes the heat makes the ground soft, allowing tectonic plates to slowly float along.
Pressure
Heat isn't enough, in and of itself, to keep the tectonic plates moving underground. Pressure also plays a major role. The heat helps keep the ground malleable while also working with pressure. Similar to how squeezing a tube of toothpaste in the middle will force the toothpaste out the front, this type of pressure also is applied to tectonic plates to keep them moving around underground, even if it normally is at a slow crawl.
Asthenosphere
The name given to the soft malleable ground that tectonic plates float on is asthenosphere. Think of this ground like soft melting plastic, so it's not natural to allow plates to move easily, but it can still be moved with enough heat and pressure. Without the softer malleable ground, the heat and pressure wouldn't matter because the tectonic plates would have no way to move.
Earthquakes
Many times tectonic plates can slide by without contact, or the contact is so light that the resulting earthquake is nothing but minor shaking or vibrations. However, when two major tectonic plates crash into one another, they can cause a gigantic earthquake. This shaking normally happens when there is irregular ground between the two plates, so instead of just slipping by one another one slips quickly and crashes into the other.