How Do We Know that the Earth revolves around the Sun?
The modern model of the Solar System is basically that of Isaac Newton. His laws of motion and universal gravitation accurately predict the motions of all the planets, asteroids, comets, natural and artificial satellites in the solar system to very high precision. This is very powerful evidence that the model is correct and the Earth, therefore, MUST go around the Sun. But is there any more direct evidence for motion of the Earth?
Yes, there is. I'll mention two pieces of evidence, abberation of starlight and stellar parallax. These share some interesting history because abberation of starlight was discovered when experiments were done to detect stellar parallax. So, what are these things?
If the Earth goes around the Sun then our postion changes throughout the year. We now know that this motion causes Earth to move, back and forth, about 186 million miles each year. If we measure the position of stars in the sky, our changing viewpoint should cause their positions to shift. How much is this shift? When the attempt to measure this was first made by James Bradley and Samuel Molyneux, in 1725-1727, it was assumed that all stars were the same brightness as the sun. The star that was chosen, Gamma Draconis, is pretty bright. About the same as the stars in the Big Dipper. Assuming it was as bright as the sun it must be really far away. It worked out to a total shift of 0.3 arcseconds.
This was just within the expected accuracy of the telescope used. The actual experimental result was more than 10 times bigger and the shift happened at the wrong time of the year! The effect was the same for all of the stars in the same part of the sky. In fact it depended only on the star's distance from the path of the Sun in the sky. This was seen as the result of the finite speed of light. The effect is called the abberation of starlight.
What about stellar parallax? It turned out the stars were further away than people had supposed and more sensitive instruments were needed to measure parallax. Finally, in 1838, Friedrich Bessel measured the parallax of a star.