Galileo believed in Copernicanism. These people believed that the Earth rotates around the motionless sun and these sorts of beliefs were frowned upon by the church during this time period. Galileo believed that church and science were separate deities. He was known as the "father of modern science."
Why was Galileo called to trial? Galileo was himself a Roman Catholic. He went to Rome to try to attempt church officials not to place on ban an his beliefs. He was told to not speak of the Earth rotating around the sun ever again, doing so would be heresy. Basically he did not listen and continued to write books about his theories and was then ordered to house arrest and lived out the rest of life there.
The reason that Galileo was brought before the Inquisition was that his ideas went against the beliefs of the Church.
Specifically, he said that the Earth moved around the Sun. In those days, everyone believed that the Sun moved around the Earth. The Church officially believed in this idea as well. This was partly because Aristotle said that that is the way the universe worked. This was also because of various things that the Bible says. The Bible says things like that the Earth was fixed and cannot move.
So because of these things, he got in trouble with the church.
Galileo was brought to trial by Inquisition twice, first time 1616 and 1633. In 1616 Galileo was summoned to Rome by the inquisition to determine if his views supporting Copernican theory that all planets including earth revolve around the sun were opposed to Church's teachings. In this trial he was cleared of the charges of heresy but was instructed to not "to hold or defend" Copernican theory.
Later, in a book published in 1632 Galileo again supported the superiority of Copernican theory over the orthodox Ptolemaic-Aristotelian theory. Once again Galileo was summoned to Rome to defend the charges of wilfully the order to not to hold or defend Copernican theory. In 1933 the inquisition declared him guilty of the charges, forced him to recant his statements, and sentenced him to life imprisonment.