Before we can answer your question, we need to clarify a few terms.

Continental drift is the theory that scientists used to describe how our modern continents were once connected into a single continent, called a supercontinent. That theory is mostly attributed to Alfred Wegener, who thought of the continents as plowing across the ocean bottoms to "drift" to their current locations. Wegener came to that conclusion by four main observations:
(1) fossils of organisms that could not have swam across oceans were found on neighboring continents;
(2) tropical plant fossils were found on Arctic continents, suggesting the continent may have been at a warmer latitude previously;
(3) the shape of the continents suggest they once fit together;
(4) mountain ranges present on different continents line up when those continents are fit back together; and
(5) when the continents are fit back together, a pattern of previous glacial erosion lines up.

Those lines of evidence are more detailed than this brief summary, but that covers the main components of that theory. A big problem with Wegener's theory is that there was no mechanism to explain why the continents would drift.

Today, that theory has been replaced by the theory of plate tectonics.

Plate tectonics describes that continents are actually 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. Today we know that 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). Depending on how the plates are moving, the crust is either being formed, destroyed, smeared against itself, or squeezed together as 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.

Ok, now for your question. There are three forces that drive plate tectonics.

(1) ridge push.
Ridge push happens at mid-ocean ridges in the bottom 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 increase more upwelling of magma to 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 and destroyed, or returned to the mantle 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 changes 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).

Continents move by plate tectonics. The crust and upper, colder part of the mantle together are referred to as the lithosphere. The lithosphere is made up of rigid plates, many of which include continental and oceanic crust. The lower, warmer part of the mantle is called the asthenosphere. Even though the asthenosphere is almost entirely solid rock, over very long timescales, it flows like play dough and convects. This allows lithospheric plates to move on top of the asthenosphere.

Mid ocean ridges are plate boundaries where the plates move away from each other, and magma comes up in between them to form new crust. The upwelling magma helps to push the plates apart. The older edges of the plates (opposite the mid ocean ridges) get denser with time, and eventually the dense edge starts to sink down into the asthenosphere and is recycled. The mantle convection and growing density of older crust allows plates to move.

Continental drift is ultimately caused by the fact that the interior of the Earth is hot and the surface is much cooler, so convection happens resulting in hot rocks coming up to the surface and colder rocks sinking downward. The continents - fluff on top of the crust - are dragged along for the ride as the ocean plates part and form at ocean ridges, and then sink down into the mantle along trenches.

It should be noted that continents do affect this cycle, too, as it is more difficult for heat to get out when it is underneath a continent. This means that continents that get too large (like Pangaea) break up when they form.