How does the Earth orbit round the Sun?

The sun moves across the sky not because it is moving but because the earth is revolving and allows us to only see the sun for about 15 hours a day.

Orbiting bodies

In a sense, the sun does orbit each of the bodies that orbits the sun. This is real, but in the case of our relatively gigantic sun, there is little if any observable effect.

Consider the earth-moon system as an example. If the earth and moon were of exactly equal mass, then observed from above you would see the two swinging around together, and you would not see a motionless earth with a massive moon revolving around it. They would in reality each be orbiting around their common center of gravity. The greater the difference in mass between the two bodies, the nearer to the larger body the center of gravity will be. In the case of the earth-moon, the center of gravity, or barycenter, is literally within the body of the earth itself.

The sun is so unimaginably massive that it is likely the barycenter between the sun and any other body of the solar system must be within the body of the sun itself. Of course, there are countless bodies orbiting the sun, so there would be some complex interactions going on among all those barycenters. But at times when the major planets are gathered together within a few degrees of one another, perhaps there would be a measurable, even if slight, effect on the position of the sun.

Binary star systems, where two suns of equal or similar mass swing around each other, are also examples of the same idea.