This is the best animation that I have seen of the ancient Earth centered system for calculating the positions of the planets in the sky. Smaller circles on the main orbits called epicycles correct for the fact that planets don't actually revolve around the Earth. It's cool to see the whole system in motion at once.
Contrary to popular opinion, there never were 'epicycles upon epicycles' needed to produce further corrections, and when Kepler finally proposed his elliptical sun centered system, he had found that the Ptolmeic model was about as accurate as the circular Earth centered Copernican model - both accurate to a few arcminutes. Another interesting fact is that the Copernican model also included small epicycles called 'epicyclets'. Both systems were mathematically quite difficult to work with.
The Copernican model was ironically praised for the fact that it adhered better to Aristotelian theory in that heavenly bodies move at a constant speed, which does not occur in Ptolemy's model - a problem that he made apologies for. Many astronomers were simply impressed that two radically different models could give accurate planetary positions.
The driving economic interest behind creating these models was the production of natal charts giving the positions of the planets at the time of birth, useful for making astrological predictions - a practice that paid the bills for the Renaissance astronomers.