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Heliocentrism, or the theory that the Earth revolves around the sun was first proposed by the ancient Greek Philosopher, Aristarchus and later developed by Nicolaus Copernicus, replacing geocentrism as the prevailing astronomical paradigm despite Christian threats of torture against its later proponents such as Galileo.
 Historical background
The geocentric model for the universe had been the common sense view of the world for many, many years. It was also the established religious dogma of the time, based on a Bible passage that described God placing the earth motionless in the cosmos. However, to compensate for the apparent retrograde motion of planets in the sky, the system became massively complicated. Copernicus was one of the first to truly challenge this idea, longing for a more simple and elegant solution. The proposal was to place the sun at the centre and have the planets orbit around it.  This makes the apparent backwards motion of the planets merely an optical illusion as the Earth races around the inside track of the solar system and overtakes the outer planets.
There were still a few problems with Copernicus' theory at the time, however. It didn't seem to explain why things fell towards the Earth and seemed to completely reject classical physics as it was described at the time. It wasn't until one century after Galileo that Kepler's observations and Newton's calculations provided even more solid evidence that demonstrated that heliocentrism is the correct model for the solar system. Two centuries later Joseph von Frauenhofer compared the spectrum of the sun to that of other stars and proved that the sun is not in a special place either.
The Earth does not orbit around the Sun following a circle, but an ellipse (the Sun is one of the two foci of the ellipse, not the center). Of course if one needs even more precision, the focus is not the center of the Sun but the barycenter of the Solar System, and the trajectory is not an ellipse but a more complicated movement influenced by other planets; actually the difference is negligible, and to be fair, Earth's influence over Sun's movement is negligible compared to the Sun's one over us.
Now we also know that this barycenter orbits the Milky Way Galaxy in about 200 million years. This movement is negligible in calculating orbits in the Solar System, because relativity proved that accelerations are absolute, but constant linear speed has no effect. The movement is circular, but the circle is so large it looks like a linear movement.
And if you really want to get technical: the Milky Way, together with the Andromeda Galaxy and an assortment of smaller galaxies in the "Local Group", orbit a common barycenter with a large number of other galaxies concentrated in the Vega Supercluster about 35 million light-years away.
If that's not good enough for you, there is dipole anisotropy in the Cosmic Microwave Background Radiation which equates to a motion of ~600 km/sec in the general direction of a gravitational anomaly dubbed "The Great Attractor". This includes the sum of the motions of the sun, the galaxy, the Local Group, and the Virgo Supercluster.