Why do tectonic plates move?

The main driving force of plate tectonics is gravity. If a plate with oceanic lithosphere meets another plate, the dense oceanic lithosphere dives beneath the other plate and sinks into the mantle. This process is called subduction. The sinking oceanic lithosphere drags the rest of the tectonic plate and this is the main cause of plate motion. Oceanic lithosphere is therefore pulled apart in several directions: that process creates the mid-ocean ridges where new, hot and light oceanic crust is created.

However, convection also drives plate tectonics. Picture this scenario: when you cook noodles in a pot of water, you create convection cells. The noodles move upward in the middle of the pot where the temperature is higher, and then downward on the edges of the pan where the temperature is lower. Such convection cells exist inside the Earth's mantle. One difference is that the mantle is not liquid; rather, the solid rocks are so hot that they can slowly flow. Hot, less dense rock material goes toward the crust whereas relatively denser, less hot material goes toward the core.

At certain times and places, hot, upflowing rock material in these convection cells weakens continental crust to create rifts and eventually new ocean basins. The East-African Rift, for instance, is the result of such a convection cell breaking up the African plate. Convection cells were responsible for the breaking up of supercontinents many times in Earth's history.

Main tectonic plates at the surface of the Earth. Subductions are where arrows point toward each other, mid-ocean ridges and rifts are where arrows diverge (Source: USGS)