What causes the Earth to move?

For objects moving with constant velocity, nothing is "causing" the object to move - there is no absolute movement, it all depends on your reference frame.

Now, if there is a change in the object's velocity (the Earth is orbiting the Sun, with its velocity is changing direction continuously), then by Newton's Second Law a force is acting upon it (the gravitational force from the Sun).

Applying this to Earth...
If you are standing still on Earth, then the Earth isn't moving relative to you (look at the ground to notice this). But if you're travelling in a car and you look at the ground through the window, you'll notice the Earth is moving relative to you (rotating, to be more precise). Which reference frame is correct? Neither! There is no preferred inertial reference frame, the physics looks the same in all of them.

Now, what I believe your question refers to is about the Earth's movement relative to the Sun.

The Earth's trajectory around the Sun is an ellipse, which in our case looks a lot like a circumference. The reason for this elliptical trajectory has to do with how gravity behaves. More precisely, the fact that the gravitational force is proportional to 1/r², where "r" is the distance to the source (in our case, the distance between the Sun and the Earth), is why we get this sort of trajectories (you can get other types of trajectories as well).

Adopting a reference frame in which the Earth is orbiting the Sun, if our star were to disappear all of sudden then the Earth would follow a straight line! There wouldn't be any source for the centripetal force responsible for the "circular" trajectory (ignoring the gravitational attraction from the other planets)! So, in this case, you can consider the Sun as the responsible for causing the Earth to move in an almost-circular trajectory, that is, with acceleration (change in velocity with time)!