Around 200 AD the astronomer Ptolemy wrote the Mathematike Syntaxis, better known now as Almagest. It was a review of the astronomical knowledge of its time. It was lost in Europe during the Middle Ages, but preserved in the Islamic world, and was finally translated back into Greek and Latin in the 16th century. The name Almagest is a Latin corruption of its Arabic name.
The Almagest detailed the geocentric worldview, which had prevailed since Aristotle: the Earth is the immobile centre of the universe. Everything above the Moon is perfect, eternal and unchanging. The bodies in space move around the Earth in perfect circular motions.
Computing the motion of the planets with the assumption that they move around the Earth and that all motion is circular gets very complicated and rather ugly quickly. Complex "epicycle" motions are needed to account for the apparent motions of the planets in the sky, which are really caused by the relative positions of the Earth and the planet changing as they move around the Sun.
Despite its problems, the Aristotelian model was philosophically appealing, especially to the Catholic Church. Catholic doctrine said that the universe is created for mankind, and having the Earth in its centre seemed proper. Also a moving Earth would have required the universe to be vastly bigger than with an immobile Earth, to explain why the stars remain static over the course of the year. The Aristotelian model was not abandoned in science until well into the 17th century.
A heliocentric system, in which the Sun is in the middle of the universe and all planets orbit around it, was not a new idea; it had been proposed by many since ancient times, but it did not really catch on because of the great authority given to the writings of Aristotle.
Copernicus developed mathematically a working heliocentric model of the solar system, but his model still had many of the problems of the old Aristotelian model. He still used perfectly circular motions, so also he had to use the complicated epicycles.
His main work on the subject was not published until the year of his death, as he was wary of provoking the Church. It was a sensitive subject; some decades after Copernicus's death, Giordano Bruno was burned at the stake, among his other heresies for supporting the Copernican model.
Tycho Brahe was a Danish astronomer, who took observational astronomy to its very limits before the invention of the telescope. He lived a very colourful and interesting life.
Tycho made possibly the best astronomical observations of his age. He worked first for the king of Denmark, and later in Prague. His contribution to the Copernican revolution is not as a theorist, though he proposed a model of the solar system, but because of his excellent observations which his student, Kepler, inherited.
Tycho also observed a supernova in the sky, and proved that it must be farther away from the Earth than the Moon, which was against the Aristotelian principle of an eternal and unchanging superlunar world.
Johannes Kepler was a brilliant mathematician and worked as Tycho's assistant. After Tycho died, Kepler, who was by then a Copernican, took Tycho's observations and eventually came up with the model of the solar system that we still consider to be true.
He showed that the observations can all be explained if the planets move in elliptical orbits (instead of circles) around the Sun, which is located at the focal point of the ellipse. Also the planets need to move faster the closer they are to the Sun, and Kepler showed the correct mathematical relationships between the distance and the orbital motion.
Kepler's results were not based in any physical theory, simply mathematical constructs to fit the observations, but within the next hundred years Isaac Newton and others developed physics and mathematics far enough that Kepler's laws could be shown to follow simply from Newton's physical principles.
After Kepler's work, the Copernican worldview finally started gaining popularity and within the next hundred of years it was widely accepted.
Though Galileo is famous as an advocate of Copernicanism, his contribution to the field is not large. Galileo was the first person to make astronomical observations with a telescope, and he was a great scientist. He made perhaps the first proper controlled scientific experiments in various fields of physics.
In his later life he published the Dialogue Concerning the Two Chief World Systems, which argued strongly for the Copernican view. The book offended the Pope, and Galileo was eventually sentenced to house arrest for the rest of his life (which turned out to be nine years).
Lenses were known well in Europe since medieval times. The first people to put two lenses together to make a telescope were in the Netherlands in about 1608.
Galileo heard of the invention soon after, and quickly built his own telescope. He was the first person to turn it towards the sky, and he observed many new astronomical phenomena: details of the surface of the Moon, sunspots and the largest moons of Jupiter.
Telescopes grew bigger and became important for astronomy in the next century or so after Galileo. Lens telescopes, especially these early ones, had several flaws, however. Large good-quality lenses were hard to make, and the size of the telescope also had to be made very large.
James Gregory first made a practical design of a telescope based on a concave metal mirror instead of glass lenses. He did not build one, however.
Isaac Newton refined Gregory's ideas and developed the "Newtonian" telescope which is still in wide use, especially by hobbyists because it is easy to construct.
Large mirrors are easier to construct than large lenses, and the form of the reflecting telescope offers other advantages over a refractor as well. These days practically all astronomical telescopes use mirrors.