This system came to be called the Ptolemaic System and was the center of astronomical beliefs for nearly a millennium and a half. It predicted the positions of the planets accurately enough for naked-eye observations.
Ptolemy described his system in his book, Almagest (Also known as Mathematical Syntaxis). It was a thirteen book mathematical explanation of astronomy, containing a wide variety of information. He also included a star catalog that contained 48 constellations, all with the same names still in use today.
The Ptolemaic System was the accepted wisdom until the Polish scholar Copernicus proposed a heliocentric view in 1543. In fairness, Ptolemy's system is actually more accurate than Copernicus's. The heliocentric calculations for the movement of planets does not improve on Ptolemy's until Kepler's Laws were added. Some people also doubt that Ptolemy truly believed his own system, rather he merely used it as a method of calculating positions.
Not just an astronomer, Ptolemy was very important in the history of geography and cartography. He was well aware that the Earth is a sphere. His is the first known projection of the sphere onto a plane. His work, "Geography" remained the principal work on the subject until the time of Columbus. It was amazingly accurate for the time, but had Asia extending much too far east. This may have been a deciding factor in Columbus's decision to sail west for the Indies.