I read a statement some time back that some people who seriously study such thilngs say that there is no way that it can be proven whether Earth moves around the Sun or whether everything moves around Earth. Sounds strange! Is there anyone out there qualified to comment on this?

Both the Sun and the Earth orbit the center of mass of the Sun-Earth system. This is ascertained by watching other orbits like the other planets or binary stars or the moon or whatever.

Actually, it is possible to construct a mathematical model of the solar system that has the earth stationary and the sun orbiting around it in an elliptical orbit and the other planets orbiting around the earth in a very complicated motion of sum of ellipses. While this works mathematically, such a solar system would not conform to Newton's laws of gravitation (which as Capuchin pointed out requires the sun and earth to orbit each other around the center of mass of the earth-sun system). If you're not worried about the physics of how gravity works, but are only interested in predicting where in the sky the planets should appear, such a system would work just fine. It's equivalent to predicting the view out your car window by assuming that the car is stationary and the rest of the world is moving and tilting as you accelerate, drive up and down hills, turn left and right, etc.

Thinking that the Earth revolves around the sun is based on the heliocentric theory of the solar system. Man was only able to prove that the Earth rotates in the 17th century when a man called Jean Foucault created what is known as the foucault pendulum. The pendulum starts always going in the same direction back and forth. However, we know the Earth rotates because over time the pendulum changes direction. But it hasn't really changed direction, the Earth has moved underneath it!
The stuff below also helps explain why the modern model is the most accepted.
I got it from: Retrograde Motion

Ptolemaic Explanation

The model of the solar system developed by Ptolemy (87 - 150 A.D.) was a refinement of Aristotle's (384 - 322 B.C.) universe. This model consisted of a series of concentric spheres, with the Earth at the center (geocentric). The motions of the Sun, Moon, and stars was based on perfect circles. To account for the observed retrograde motion of the planets, it was necessary to resort to a system of epicycles, whereby the planets moved around small circular paths that in turn moved around larger circular orbits around the Earth. This accounts for retrograde motion, as shown in the animation below:


Copernican Explanation

Copernicus replaced the geocentric universe of Ptolemy with one that was centered on the Sun (heliocentric), with only the Moon orbiting the Earth. His model was still based on circular orbits (and therefore still required further refinement), but it was able to achieve superior precision than the Ptolemaic model without the need for epicycles or other complications. The explanation for retrograde motion in this system arises from the fact that the planets further from the sun are moving more slowly in their orbits than those closer to the sun. The retrograde motion of Mars occurs when the Earth passes by the slower moving Mars, as shown in the following animation:

Yes The Earth Revolvs Around The Sun. It Sounds Strange But It Was Believed By The Proofs Given By The Astronauts