Ptolemy definitely made astronomical observations from Alexandria in Egypt during the years from 127 to 141. We suspect his teacher was Theon of Smyrna, who was both an observer and a mathematician who had written on astronomical topics such as conjunctions, eclipses, occultations and transits.
The earliest, and perhaps most important of Ptolemy's work that has survived is the Almagest, a treatise in 13 books. It gives in detail the mathematical theory of the motions of the Sun, Moon, and planets. Ptolemy made his most original contribution by presenting details for the motions of each of the planets. His theories were not superseded until a century after Copernicus presented his heliocentric theory in 1543.
Ptolemy first of all justifies his description of the universe based on the earth-centered system described by Aristotle. It is a view of the world based on a fixed earth around which the sphere of the fixed stars rotates every day, this carrying with it the spheres of the sun, moon, and planets. Ptolemy used geometric models to predict the positions of the sun, moon, and planets, using combinations of circular motion known as epicycles. Having set up this model, Ptolemy then goes on to describe the mathematics which he needs in the rest of the work. In particular he introduces trigonometrical methods based on the chord function.
Because of systematic errors in his works, Ptolemy is accused by some of just correcting Hipparchus' data for precession. Indeed there is some evidence that Ptolemy is guilty of stealing and fabricating data. But he is still one of the most influential Greek astronomers and geographers of his time.