To anyone who stands and looks at the sky, it is apparent that the earth under one's feet stays in one place while everything in the sky goes around once every day. Observing over a longer time, one sees more complicated movements. The Sun makes a slower circle over the course of a year; the planets have similar motions, but they sometimes turn around and move in the reverse direction for a while (retrograde motion). As these motions became better understood, they required more and more elaborate descriptions, the best of which was the Ptolemaic system, formulated in the 2nd century.
The strange idea that it is the other way around—a heliocentric theory—was first proposed by Aristarchus (c.270 BC). When he wrote, the size of the Earth had been calculated accurately, and he himself measured the size and distance of the Moon and Sun; his figures were not accurate by modern standards, but a serious start. Perhaps, as many people have suggested, paying attention to these numbers led him to think that it made more sense for the Earth to be moving than for the huge Sun to be moving around it. Some people in his own time, though, considered the idea to be against religion.
In the 16th century the theory was revived by Nicolaus Copernicus, in a form consistent with then-current observations. This theory solved the issue of planetary retrograde motion by arguing that such motion was only perceived and apparent, rather than real: it was a parallax effect, as a car that one is passing seems to move backwards against the horizon. This issue was also resolved in the geocentric Tychonian system; the latter, however, while eliminating the major epicycles, retained as a physical reality the irregular back-and-forth motion of the planets, which Kepler characterized as a "pretzel".
Heliocentrism was notably advanced by Galileo, Kepler, and Newton. It was vigorously resisted, though, by elements in the Catholic church, who prevailed in showdowns in 1616 and 1633 and officially suppressed heliocentrism. The favored system had been that of Ptolemy, in which the Earth was the center of the universe and all celestial bodies orbited it. (The Catholic support for geocentricism should not be confused with the idea of a flat earth, which the Church never supported.) When prominent Catholic astronomers, including Clavius, became dissatisfied with the Ptolemaic system, many moved to the rival Tychonian system, a geocentric compromise; after 1633, the use of this system was almost mandatory. For advancing heliocentric theory Galileo was put under house arrest for the last several years of his life.
One objection against Heliocentrism was that, if it were true, parallax should be observed in the fixed stars. Simple geometry would show that the parallax would be too small to detect if the stars were sufficiently far away; but that implied a universe so large that it was considered incredible. Stellar parallax was finally observed with the greatly improved instruments of the 19th century, by which time the truth of heliocentrism in a very big universe was no surprise to anyone.
The strictly heliocentric view was abandoned in steps. That the Sun was not the center of the universe, but one of innumerable stars, was strongly advocated by the mystic Giordano Bruno; Galileo made the same point, but said very little on the matter, perhaps not wishing to be burned at the stake. Over the course of the 18th and 19th centuries, the status of the Sun as merely one star among many became increasingly obvious. By the 20th century, even before the discovery that there are many galaxies, it was not an issue.
As to the center of the Solar system, the history is more complex. Kepler and Newton both demonstrated that the Sun was not strictly at the center, but at one focus of each planet's elliptical orbit. In fact, the Earth was not strictly at the center even in the Ptolemaic system. The 20th century produced a more radical change: technically, the Theory of relativity made the heliocentric model obsolete, because no object can be definitively described as orbiting another or as being orbited. Objects can only be described as being mutually in relative motion.
There is an obvious, and incorrect, objection to the idea that any object can equally well be taken as the center of the world: If the Earth stands still, then the whole universe is spinning around at speeds up to billions of light-years per day, and we know that nothing can travel faster than light. In the mathematics of the General theory of relativity, however, the objection is taken care of.
In modern terms, then, neither the geocentric nor the heliocentric model is truer than the other. "The two theories...are physically equivalent to one another."  However, for practical purposes, it remains the case that describing the sun as the (approximate) center of the planets' orbits is much simpler mathematically and more intuitively useful.
- Sir Fred Hoyle, Nicolaus Copernicus, 1973.
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