The Inquisition was a Catholic Church tribunal that existed to protect the Church from heresy.  There were many inquisitorial tribunals in different nations and at different times beginning in the 13th century.  Some of these were more powerful than others, but all inspired fear in the hearts of those called to appear before them. 

During the late 16th century and into the 17th, the Church was in a battle for survival.  Protestantism was spreading throughout Europe and the Church, believing that it had to defend its primary theological precepts against the heresy of Protestantism, instituted another version of the Inquisition.  It was before this tribunal that Galileo was called to defend himself against heresy, a charge brought against him because of his teaching that  the earth revolved around the sun and that the sun, not the earth, was the center of the universe (heliocentric theory).   Copernicus had said that about 100 years before.  His theory had been condemned by the Church as heretical since it seemed to contradict the Bible and, therefore, Church teaching which said that the earth was the center of the universe (geocentric theory).

However, contrary to what most people think, the Church did not actually oppose the science behind Galileo's theory of a sun-centered universe.  In fact, quite a few of those Churchmen who sat in judgement of Galileo were themselves scientists.  What the Church was afraid of was that Galileo's theory would lead to an individual interpretation of Scripture and that was something the Church feared.  Remember that the Church was at this time involved in fighting and defeating the Protestant movement spreading throughout Europe.  Galileo, therefore, had to be stopped at all cost!