Before Copernicus, everyone thought that the earth was immobile at the center of the universe, a concept called "geocentrism". Their grounds for this were pretty simple: we don't seem to be moving and everything else does; and everything else seems to be moving around us. Since people thought that everything revolves around us, they concluded that ... well ... everything revolves around us; that is, we are of central importance. This fits well with the biblical narrative, which makes the audacious claim that we have an intrinsic significance and value. In Pale Blue Dot, Carl Sagan, obviously not a proponent of this position, represented its egotism well:
(I)f the lights in the sky rise and set around us, isn't it evident that we're at the centre of the Universe? These celestial bodies ... circle us like courtiers fawning on a king. Even if we had not already guessed, the most elementary examination of the heavens reveals that we are special ...
But then Copernicus discovered that the earth revolves around the Sun. This "revelation" dethroned humanity by removing it from its central position; and because of this, the Catholic church denounced it. This is one of the primary examples of how science conflicts with, or simply refutes, religion. There's a small problem with it though: it's mostly bunk. Let's investigate it, shall we?
First, and least controversial, is that the geocentric model of the universe did not come from the Bible but from Aristotle's cosmology, which was later adopted and made rigorous by Ptolemy. According to this model, the universe was arranged in concentric spheres, with God, the Prime Mover, on the outside keeping it in motion. This was the model throughout the Middle Ages, and Thomas Aquinas explicitly endorsed Aristotle's philosophy in the 13th century. Later, when the Catholic church sanctioned Aquinas' philosophy, this included his sanction of Aristotle.
Second, the most vociferous denunciations of heliocentrism were not made by the church but by the scientific establishment of the time. They had, after all, devoted much of their scholarship towards expounding upon the geocentric model, and were then being told that their paradigm was wrong and their lifes' work meaningless.
Third, the Catholic church did of course denounce Galileo over his writings on this issue, but the controversy wasn't as two-dimensional as is commonly portrayed. For one thing, Galileo had publicly mocked Pope Urban VIII who, until that point, had been a personal friend. The pope misused his power to call Galileo to account, and the heliocentric theory was the excuse he used. Thus, it was more of a political persecution than a religious one. Another issue was that Galileo had claimed that the Catholic clergy had erred in their theology and their interpretation of Scripture by accepting geocentrism. This wasn't long after the Protestant Reformation, so they were a little put off by this. In particular, the texts in question give no indication that they are intended to be understood from a perspective other than the surface of the earth.
But probably the biggest error in this metanarrative is the equation of geocentrism with anthropocentrism. That is, since people thought we were centrally located, it implies that they thought we are centrally significant; that since we are in the center of the universe, we must be the most important thing in it.
This was emphatically not the case. In the ancient/medieval cosmology, the closer one was to the center indicated one's lack of significance and value, that one was less esteemed and privileged. Aristotle had argued that the universe is a sphere which the Prime Mover kept in motion from the outside. So what's the furthest place within a sphere that is furthest from what is outside it? The center. Thus, the center was the place in the universe furthest removed from God (although this does not seem consonant with the Christian concept of God's omnipresence). That they thought the earth valueless is further illustrated by the fact that in this cosmology, the heavenly objects beyond the sphere of the Moon were made up of a fifth element, the quintessence, which was not prone to corruption like the other four elements that the earth consisted of (earth, air, fire, and water).
The medieval theologians tried to account for geocentrism in their hamartiology (doctrine of sin). What was sinful was heavy, and fell towards the center, whereas the heavens were holy and perfect. Essentially, the earth was the universe's toilet. The heaviness of sin also made the earth immobile, as distinct from "the dance of the heavens". Motion was a good thing; the fact that the earth didn't move showed that it wasn't good.
This is easily illustrated. In their view the earth was at the center of the universe, but what was at the center of the earth? Hell. Thus the inhabitants of hell were even closer to the center than the inhabitants of the surface of the earth. Did the medieval theologians think this made hell a place of esteem and its inhabitants more valuable? Of course not. In fact, as Dante wrote, at the very center of hell was Satan, bound and immobile. So, as Arthur Lovejoy put it in The Great Chain of Being, the medieval cosmology is better described as diabolocentric than geocentric.
Thus, the claim that the earth is not at the center of the universe was a huge promotion for humanity, not a demotion. Galileo clearly understood this when he wrote that the earth "is not the sump where the universe's filth and ephemera collect." It wasn't until the mid-17th century that some French satirists first suggested the popular story line that Copernicus' discovery represented a demotion for humanity. C. S. Lewis, in The Discarded Image, explains why the claim that the earth is at the literal, physical center does not entail that it is at the metaphorical, existential center as well.
Because, as Dante was to say more clearly than anyone else, the spatial order is the opposite of the spiritual, and the material cosmos mirrors, hence reverses, the reality, so that what is truly the rim seems to us the hub. ... We watch "the spectacle of the celestial dance" from its outskirts. Our highest privilege is to imitate it in such measure as we can. The medieval Model is, if we may use the word, anthropo-peripheral. We are creatures of the Margin.
Much of this information is addressed in greater detail in several essays by Dennis Danielson, a literary historian at the University of British Columbia. One that can be downloaded online is "Copernicus and the Tale of the Pale Blue Dot"; his description there of how scientists have responded to his arguments is especially interesting. Another Danielson essay is "The Great Copernican Cliché", published in The American Journal of Physics 69/10 (2001): 1029-35. He also wrote "Myth 6: That Copernicanism Demoted Humans from the Center of the Cosmos" in Galileo Goes to Jail and Other Myths about Science and Religion, edited by Ronald Numbers. Danielson also addresses this point briefly in his biography of Rheticus entitled The First Copernican. I quoted the relevant text here.
Two more points: first, one of the main areas of research in several scientific disciplines today is the Anthropic Principle. This is the idea that in order for life to exist (and especially advanced life), the universe has to have very specific properties. That is, if various aspects of the universe were any different, it would preclude the possibility of life existing anywhere at any time in the history of the universe. There are literally dozens of such properties that have to be precise to absurdly fine degrees, and more are being discovered just about every month.
The claim that the universe is "fine-tuned" has some obvious religious implications: if the universe is exactly the way it has to be in order for us to exist, it suggests that someone rigged it precisely for this purpose. One of the most common objections to this has been that it flies in the face of the whole history of science, which has repeatedly demonstrated that humanity has no significance. This objection is derived first and foremost from the Copernican demotion of humanity. If this paradigm is false, one of the primary objections to the Anthropic Principle is based on a misunderstanding of the history of science and religion.
Second, when people are told that their religion leads to absurd beliefs like geocentrism, some respond by essentially saying "Really? That's what we believe? Well then, let's defend that position!" Today, there is a Christian ministry that defends "geocentricity" (they think this term doesn't have the historical baggage that "geocentrism" does). Additionally, some young earth creationist ministries have jumped on board and are defending a broader concept: galactocentrism (that our galaxy, the Milky Way, is at the center of the universe). Both of these efforts are motivated by the claim that if we are centrally important, we should expect to be centrally located as well. But this claim comes from opponents of Christianity trying to mock it. I suggest that we should not let those who deprecate our faith define it for us.