Galileo's (1564-1642) championing of heliocentrism (the Sun at the center of the Galaxy) was controversial within his lifetime, a time when most subscribed to geocentrism (the Earth at the center of the Universe). He met with opposition from astronomers, who doubted heliocentrism due to the absence of an observed stellar parallax (the apparent change in position of the stars due to orbiting the sun). He also met with religious opposition based on the current scientific understanding of his time. 

It is a complex topic and so some points about attitudes at the time are important. The Catholic Church had no problem with geocentrism provided the scientific proof can be shown. For instance at the time Cardinal Baronius said that the bible "is intended to teach us how to go to heaven, not how the heavens go." And he pointed out correctly that both St. Augustine and St. Thomas Aquinas taught that the sacred writers in no way meant to teach a system of astronomy. St. Augustine wrote, "One does not read in the Gospel that the Lord said: I will send you the Paraclete who will teach you about the course of the sun and moon. For He willed to make them Christians, not mathematicians." Quotes from The Galileo Affair. (We can add today that the Paraclete (the Holy Spirit) also was not sent to teach us about dinosaurs.) 

Cardinal Bellarmine was sent to talk with Galileo about this. When Cardinal Bellarmine met with Galileo he said, "While experience tells us plainly that the earth is standing still, if there were a real proof that the sun is in the center of the universe…and that the sun goes not go round the earth but the earth round the sun, then we should have to proceed with great circumspection in explaining passages of scripture which appear to teach the contrary, and rather admit that we did not understand them than declare an opinion to be false which is proved to be true. But this is not a thing to be done in haste, and as for myself, I shall not believe that there are such proofs until they are shown to me." (Letter to Foscarini , 12 April 1615) 

At the crucial trial of Galileo about teaching heresy, Galileo claimed three proofs for geocentrism all of which don't prove geocentrism. That the tides are due to the change of speed on the surface of the earth much like a cha-cha amusement ride, the movement of sun spots and the phases of venus. Tides are due primarily to the moon, sun and coastal and sea floor shapes and some minor factors, not the movement of the earth itself. Sun spots can be explained with the rotation of the sun, not that the earth goes around the sun. The last point of the phases of venus is a more serious point but not sufficient on its own. There are more details about these proofs here. What brought conclusive proof was stellar aberration by James Bradley in 1729 which eventually led to the Church formally lifting its ban on Galileo’s books in 1758. That's a long time after 1615.

Scientifically, Galileo did not have proof this point is absolutely clear but what about the Church medling with science itself. Why did the Church step in to condemn Galileo? As what has already been hinted at, scipture has many references to the sun going around the earth. It seems to be obvious to anyone on earth. They see it moving. So without proof it appears Galileo is going against scripture and that is what he was tried for. There are many other historical issues about the trial such as Galileo having a strong ego that got in the way of his previous friendship with the current pope. Clearly the careful analysis of the case itself was primarily about science which you can see here. 

There are many inaccuracies about Galileo's trial. He did not stamp his feet and cry out, "But it does move." He was submissive throughout the trial. He was not tortured. His house arrest for the last nine years of his life which were lived in a palace where he could freely work on his research provided he did not teach about geocentrism. He actually stayed in three places, the palace of the Grand Dukes Ambassador, Niccolini, at Rome; the residence of his friend Archbishop Piccolomini at Venice, and his own villa of Arcetri near Florence. He completed two more works during this time. 

So it appears Galileo was wrong with his evidence. If Galileo had pulled his head in earlier there would have been no problem, ie if he hadn't said he had proof for geocentrism. If the Church had not pushed ahead with trying him for heresy there would not have been this issue. 

Yet, Galileo accepted the inerrancy of Scripture. So what can we put this condemnation down to.  Bertrand Conway explains it well in Galileo Galilei:

In a word the condemnation of Galileo was due chiefly to the vast majority of the theologians of the day who urged the Holy See to condemn a scientist who attacked their favorite Aristotle, and ventured to teach them how to interpret the Scriptures.
Their pride had been hurt by the many bitter attacks of Galileo, who had in private conversations and public print styled them stupid and ignorant. Their dread of the private interpretation of the Scriptures — so harmful in Germany and England — had made them overcautious. Their egregious blunder consisted in branding a scientific theory as heretical, when at best they should have considered it as unproved. They were right in maintaining the general law of exegesis that Biblical texts are to be taken in their literal sense, unless there exist good reasons to the contrary. They were wrong in forgetting the wise teaching of St. Augustine and St. Thomas that in describing the phenomena of nature the Bible speaks according to appearances.
In a sense the condemnation of Galileo was providential. It proved for all time that fallible bodies like the Roman Congregation ought not to dub a scientific theory heretical, and it prevented them from making a similar mistake for over three centuries. It proved also that whenever there is apparent contradiction between the truths of science and the truths of faith, either the scientist is wrong in advancing a mere hypothesis as a fact, or that the theologian errs in mistaking his personal opinions for the teaching of the Gospel.
Pastor concludes his brief but able study of the case with the wise words: There has been no second Galileo case (History of the Popes. Vol. XXIX., p. 62).

The Catholic Church has publically apologised for the Galileo affair. On 31 October 1992, Pope John Paul II expressed regret for how the Galileo affair was handled, and issued a declaration acknowledging the errors committed by the Catholic Church tribunal that judged the scientific positions of Galileo Galilei, as the result of a study conducted by the Pontifical Council for Culture.

But while the condemnation was an official act of the Church, it was by a fallible court that has no authority to make official declarations on faith and morals (the magisterium). So a Church court made a mistake, which the official Church (John Paul II) has publically apologised for. The mistake has never been repeated, yet it is a crucial argument for justifying the conflict between science and religion. Such a conflict is absurd when the evidence is examined. Anyone that uses such an argument should be explained the facts and invited personally to accept the official apology of the Church. 

One final point from this era is that Galileo was not the only scientist of his day. 26 of the 35 Jesuit scientists who have lunar moons named after them were alive during the life of Galileo. You might like to check out their achievements or even the list of Catholic scientists.

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