First things first, it seems a clarification of what an inquisition was is necessary considering your capitalization of the word. There was no singular "the Inquisition." There were many of them such as the Spanish Inquisition, Portugese Inquisition, or Roman Inquisition. Inquisitions were a judicial branch of the Catholic Church up until 1965 that were responsible for a variety of functions, but typically made appearances in response to a particular heresy that needed to be addressed in a particular geographic area. (Fun fact: inquisitions still exist today, but instead it is called the Congregation for the Doctrine of Faith) For example, during the time of the Spanish Inquisition there was a belief that there were many "fake Christians" due to the mass conversions of Jews and Muslims in Spain due to the nationalistic fervor of the Reconquista, economic jealousy, and upper class unrest. I've written a thesis on the Spanish Inquisition, so I know I'm over simplifying this but essentially there needed to be a special method to find out whether someone was guilty of faking Christianity. Government courts were not equipped or trained to know all the ins and outs of Church law, so special inquisition tribunals had to be set up to deal with this. If you have any more questions about the general nature of inquisitions, please feel free to ask me! My apologies if I have assumed incorrectly about your personal understanding of inquisitions.

As far as Galileo's case, he was tried by the Roman Inquisition. Unfortunately, my area of expertise is not in the Roman Inquisition but I do have an extensive knowledge of Galileo's trial. First, I'd like to clear up that the Church did not hate Galileo or anything of the sort. In fact, he was good friends with many cardinals and even the pope who ended up overseeing that Galileo was tried by an inquisition tribunal. Once, he had even been invited to a feast in his honor in Rome to celebrate his scientific achievements.

Regardless of this, Galileo had been causing a fair amount of controversy with his heliocentric theories that had been discussed in Sidereal Messenger (1610), Letter to the Grand Duchess Christina (1615), and most of all, Dialogue on the Two World Systems (1632). While his conclusions themselves did garner a bit of controversy for a variety of reasons, such as reaching different conclusions than a group of respected Jesuit astronomers, what established the most controversy was his participation in theological debates. Some Biblical scholars of the time felt that scripture did not support Galileo's theories and called him out on it openly. Despite being warned several times not to do so, he would try to twist Biblical quotes to fit his science. Since the Protestant Reformation was still fresh in the mind of Catholic theologians, it didn't sit well that a non-expert on the subject was interpreting passages differently.

In 1616, the Holy Office legally forbid Galileo from teaching his heliocentric theories as fact. However, Cardinal Robert Bellarmine issued him a certificate that allowed to him to study and research his theories further. In 1623, Pope Urban VIII gave Galileo permission to publish another work on heliocentrism, but again reminded Galileo of his legal obligation to not present his arguments as fact.

Galileo did not listen and felt his arguments had more than enough evidence to be presented factually. While Galileo was largely correct in the long run with his theories, it is important to note that other leading astronomers, such as Tycho Brahe, of the time did not believe there was enough evidence yet. They were most largely concerned with something known as a "parallax shift" that Aristotle had also had an issue with ,many years earlier. In addition, when presenting arguments against his theories in his Dialogue on World Systems he put arguments that Pope Urban had given him against his work in the mouth of a character named "Simplicio." He also insulted a leading Jesuit astronomer.

In 1633, Galileo was officially summoned for trial under Fr. Vincenzo Maculano da Firenzuola. He was arrested for breaking his legal obligation and personal oaths from 1616 to not teach his hypothesis as fact. However, it is important to notice that like the legal system today there was a fairly great deal of controversy over what Galileo should be officially arrested/tried/sentenced for. Different people involved in the trial had different interpretations.

Just for additional information about Galileo's treatment during this time, we have a mixed bag. Galileo was very ill during this time and despite a letter from his doctor to the inquisition, he was still forced to appear in Rome. However, Galileo was given private servants and chefs and put up in the extravagant Medici Villa while in Rome. When Galileo was forced to recant his theories and put on house arrest, he was allowed to stay in his Villa in Florence and was allowed to visit his daughter in a nearby convent on occasion. His friend Nicolini also was allowed to regularly visit him. Near the end of his life while old and blind, Galileo was allowed to publish another scientific work. There is a common myth that Galileo was physically tortured, but there is no proof that this ever occurred.

Some good reading I'd recommend includes:

The New Geocentrists by Karl Keating

The Catholic Church and Science: Answering The Questions, Exposing the Myths by Benjamin Wiker

The Trial of Galileo, 1612-1633 by Thomas Frederick Mayer (I can't recommend this one enough, it has translated transcripts directly from Galileo's trial within it)