Earth's crust moves by plate tectonics.

Earth's crust can be continental (the ground humans live on) or oceanic (the rock that forms at the bottom of the ocean at mid-ocean ridges).

The crust is resting on top of thick slabs of rock, called tectonic plates. These plates are always moving and interacting with other plates; this process is called plate tectonics.

Tectonic plates can move because the rocks below the plates have the ability to flow even though they are still solid and not liquid. You can think of this movement in the same way taffy can flow, even though it is still a solid. Depending on how the plates are moving, the crust is either being formed, destroyed, smeared against itself, or squeezed together into thicker crust. These motions explain all of Earth's surface geology from where oceans and mountains form, why and where earthquakes and volcanoes exist, and so much more.

There are three forces that drive plate tectonics.:
(1) Ridge push. Ridge push happens at mid-ocean ridges, at the center of the ocean bottoms, due to gravitational forces acting on the spreading ridges. Newly formed oceanic crust is less dense than old oceanic crust that has cooled and become covered in sediment. Therefore the buoyancy of the new crust can enable more magma to rise and push the ridges apart.
(2) Slab pull. Slab pull happens at the edges of ocean basins where oceanic crust is being destroyed under the edges of continents due to gravitational forces in the subduction zones. Subduction zones are where the oceanic crust meets the continental crust. Because oceanic is thinner and denser than continental crust, the oceanic crust is pushed under the continental crust where is sinks below Earth's crust. As the oceanic crust is pushed under, gravity can pull the crust faster or slower depending on its density. The faster the slab is pulled down, the faster the ridge will be pulled apart, the faster new crust is created at the ridge.

These are the main drivers behind how Earth's crust moves around. However, plate movements are not that simple. Sometimes ocean crust runs into ocean crust, continental crust runs into continental crust, the plates never collide but grind past each other, or the plates tear apart. These other settings complicate plate motion and explain how we know Earth's continents not only were once all connected as a supercontinent, but have actually gone through several cycles of smashing together and ripping apart. The reason gravity can push and pull slabs is due to the final driving force:
(3) convection in the mantle. Mantle convection is largely heat driven and explains how small perturbations in the mantle can change how the plates are moving relative to other plates and where that movement happens (for example, why continents rip apart to eventually form new ocean basins or why the Himalayan Mountains continue to rise higher and higher ).

You may have seen convection happen when boiling water. The hot water touching the bottom of the pot rises to the top, while the cooler water from the top of the pot flows in to replace the rising hot water. While boiling, the water is convecting around in the pot.

The interior of the Earth is hot. Hot material is less dense, so rises to the surface. This means that the cold crust on the surface has to move out of the way as the hot mantle comes up underneath it.

Earth's crust sits on soft stuff underneath, like a pie crust. The soft stuff underneath moves, and the pieces of Earth's crust move with it and bump into each other or move away from each other.

Earth's crust moves because new crust is constantly being produced at mid ocean ridges, where magma comes up to the ocean floor, pushing the existing crust away from the ridge.

The older crust far from the ridge is colder, which makes it denser, so it sinks down, and drags the rest of the crust behind it. The combination of new crust being created and old crust sinking makes all of it move.