How does the force of density affect tectonic plates?

The force of density, more specifically the force due to density differences is what drives tectonic plates. Density difference throughout the mantle is due to variations in its temperature. Warm areas expand, become less dense and rise while cold areas sink.

For example, continental plates are thick and light but its material erodes. When it becomes sea-floor sediment, its density increases and is then subducted with the oceanic plate.

Tectonic plates are able to move because the Earth's lithosphere has a higher strength and lower density than the underlying asthenosphere. Lateral density variations in the mantle result in convection. Their movement is thought to be driven by a combination of the motion of seafloor away from the spreading ridge (due to variations in topography and density of the crust that result in differences in gravitational forces) and drag, downward suction, at the subduction zones. A different explanation lies in different forces generated by the rotation of the globe and tidal forces of the Sun and the Moon