The Earth's tectonic plates move because of the way lava moves inside the planet. The center of the Earth is a ball of molten metal. Outside of that molten metal is a layer of lava that moves the plates as it heats and cools beneath the Earth's crust.
The molten ball of metal at the center of the Earth is the core, and it heats the lava between the core and the Earth's crust. The lava closest to the core moves toward the surface of the Earth as it is pushed aside by cooler, denser lava. The lava cools as it reaches the tectonic plates between it and the surface. Most of the cooler lava is too heavy to break through the crust and sinks back down toward the core. This motion creates a convection cell beneath the Earth's tectonic plates and causes the plates to move.
There are several large tectonic plates and innumerable smaller ones that are shifted by the underground convection cells. The plates are all pressed closely together, and when one plate is forced beneath another, it forms mountains. Plates moving away from each other create rifts, and breaks in the Earth's crust allow lava to flow out.