This is a really interesting question, because the practice of astronomy is, as you know, old enough to go back to the Greeks (and even before them!). In fact, many civilizations across time have looked up to the sky and struggled to understand how it all worked. But I'll limit the scope of what I'll tell you just to the Western tradition (the Ancient Greeks and Europeans).
In the times of Antiquity, the dominant model of the Solar System was the Ptolemaic model, named after the Hellenic astronomer Ptolemy. This model put the Earth at the center of the Universe, and the Moon, the Sun, and each planet revolved in a circle around the Earth on what was thought of as their own "celestial sphere," or sometimes called a "crystal sphere." Imagine that each planet is embedded in a transparent shell of glass, and the whole sphere rotates around the Earth. (Some models from the Middle Ages, however, say that there's a designated angel assigned to each planet to push it along its orbit!) Anyway, beyond the planetary spheres lies the sphere of the "fixed stars," which is like the sphere of the planets, but this crystal sphere is speckled with hundreds of thousands of all the stars that the Ancients could see with their naked eye. It's important to understand that this sphere lay just outside the sphere of Saturn (which was the farthest planet that they could see with their naked eye; Uranus wouldn't be discovered until 1781 with the help of the telescope). The Greeks didn't understand that the Sun was actually a star, because the stars are so far away they appear as little points of light, hardly anything like the Sun.
Diagram of the Ptolemaic Model: http://history.wisc.edu/sommerville/351/351images/ptolemaic.gif
During the Middle Ages, scientific development pretty much stopped. The opinions about how the world worked hardly changed at all since the time of the Ancient Greeks; philosophers, thinkers, and theologians of the Middle Ages pretty much took the words of Aristotle, Ptolemy, etc. to be absolutely true and basically perfect already.
However, once the Renaissance flowered up in Italy and spread to the rest of Europe, a huge wave of discoveries in astronomy and physics challenged the basic ideas held by the Ptolemaic model:
1) Copernicus revived the idea of heliocentrism, the idea that the Sun is the center of the Solar System, and not the Earth. This idea had been around for quite some time, and its invention is usually attributed to the Greek astronomer Aristarchus, but it was commonly rejected. Plutarch reported that Cleanthes, a contemporary of Aristarchus, said, "it was the duty of the Greeks to indict Aristarchus on the charge of impiety for putting in motion the Hearth of the Universe [the Earth]… supposing the heaven to remain at rest and the earth to revolve in an oblique circle, while it rotates, at the same time, about its own axis." (Russell, Bertrand. "History of Western Philosophy")
2) Later, with the help of observational data from Tycho Brahe, Johannes Kepler developed his three laws of planetary motion, which said that the Sun was the center, and the planets moved around it in ellipses, not circles. This was an especially annoying idea to those who held to the classical Greek model, because circles were thought to be "perfect," and why wouldn't the heavens be perfect?
3) Galileo Galilei used his improved telescopes to see that Jupiter has four big moons (we know now that Jupiter in fact has 63 moons, but these are the biggest ones that Galileo could easily see, which is why we call them the "Galilean Moons"). This also upset people who held on to the Ptolemaic model, because things should all be orbiting the Earth! It's imperfect for these moons to be orbiting something else! Galileo also found that, with the help of his telescope, that Venus goes through light and dark phases like those of our Moon. The only explanation for this is that Venus goes around the Sun in an orbit between the Sun and the Earth.
4) Isaac Newton used Kepler's laws together with his Universal Law of Gravitation explained that the planets moved around like they do because of the gravitational attraction of the Sun, and not angels pushing them around.
5) One of the final nails in the coffin on the Ptolemaic model came from the English astronomer duo of Edmond Halley (the same one we name the comet after) and John Flamsteed. These two worked on a newer and more accurate catalog of the stars using improved telescopes from the ones Tycho Brahe used. When Halley compared the new catalog to an older catalog compiled by Hipparchus in second century B.C., he found that, although most of the stars hadn't really moved at all, some stars had moved dramatically since ancient times. Imagine their surprise when they found that a big bright star had moved the equivalent of two widths of a full moon across the sky over 2000 years! Since some of the stars had moved, but some of them had not, this suggested that, not only were the stars really far away, but we were looking at them from different distance; they couldn't possibly lie in the same solid crystal sphere if they all move independently! This was evidence that the stars move relative to each other in three dimensions.
And this was all before 1800! With the invention of more powerful telescopes, we discovered Neptune, Pluto, and the countless moons in our Solar System. Orbiting space telescopes like Hubble helped us look farther than anybody ever had. We found comets and asteroids, binary star systems, nebulae, galaxies, and black holes. The recent arrival of the Cassini space probe in the moon systems of Saturn have brought back remarkable photos of Saturn's many and diverse moons in clarity that we've never had before. As we speak, the Kepler Extrasolar Planets project is discovering planets in other star systems! Einstein's theory of general relativity has helped us try to understand how the entire Universe works. I could go on forever about the vast diversity of stuff that we've seen everywhere in space, but I think you get the idea.
In sum, I can answer your question by saying that our current understanding of the Solar System, and the Universe in general, has moved us gradually away from center stage. At first, we were at the middle of everything, fixed, and everything literally revolved around us perfectly. Then we found that the Sun is actually the center, and things aren't so "perfect" in the Solar System. We found that we're actually really far from the center of our own galaxy, around which the Sun actually orbits. Then take into account that our galaxy is one of billions of billions of galaxies in the Universe. (There are even some theories that say there's billions and billions of other universes!) All in all, we live in a bustling, diverse, energetic, and exciting Universe, and not so much a perfectly designed glass bubble floating in the middle of the "fixed stars."
Hope this helps, I sure had fun writing it!
Gribbin, John. "The Scientists." 2002 Penguin Books.
(published on 04/11/2011)