Spain had a slightly different situation than much of Western Europe, because historically it had a large Muslim and Jewish population. Ferdinand and Isabella would eventually release the Alhambra Decree in 1492 (yes, the same year Isabella financed Columbus' travels), requiring all non-Christians to convert or leave the country on pain of death. There were two terms for these converts. Conversos were Jews who converted to Christianity. Moriscos, from the word for "Moorish," were converted Muslims.
Prior to 1492, however, there was still a desire to ensure that converts from Judaism and Islam to Christianity were sincere, but after 1492 the job of the Spanish Inquisition became more widespread and intense, since so many who could not afford to leave the country converted under duress.
Besides heresy and apostasy ("backsliding" to your previous beliefs), the Spanish Inquisition went after witchcraft, blasphemy, bigamy, sodomy, Protestantism (when it came along later), and even Christian mystics (called alumbrados, "illuminati").
It was a Dominican, Fray Alonso de Ojeda, who convinced Isabella that there were "Crypto-Jews" in Andalusia: Jews who had converted but practiced Judaism in secret. This was confirmed by Isabella and Ferdinand's confessor, Tomás de Torquemada (a name that became synonymous with torture). The Spanish Inquisition held its first investigations in late 1480, and had a result by 6 February 1481, when six people were burned alive in public, a practice known as auto da fé, a Portuguese phrase meaning "act of faith." (The illustration shows an auto da fé in the public marketplace.)
Not all subjects were executed. Public flogging, imprisonment, and exile were also used, as well as being force to serve as a galley-slave, forced to row on the royal ships. There were also monetary fines. "Surviving" the investigation did not mean resuming life as you knew it: you and your descendants were prohibited from certain high-level professions like doctor and tax-collector.
If you were a good and faithful converso or a morisco, however, you were probably safe, right? Hmm, tomorrow let's see how converts were actually treated by society.
(By the way, everybody expected the Spanish Inquisition! The Inquisition protocol was to send you a message saying to be available for interrogation.)
