You could say (if you choose to look at the business with an outsider's eye) that the Society of Jesus was intended to make up for the flaws of and clean up the mess left by the Spanish Inquisition.
Let's start a bit further back; with the papal inquisition of the late twelfth/early thirteenth centuries. This had largely been a response to the surge of new ideas as a result of the 'little renaissance' that was the first fruits of the plunder of the Crusades- mostly graeco-roman rediscovery. Orthodoxy suddenly became an issue in a way that it had not really been since the fifth century.
The problem from the church's point of view being that these early inquisitors were curious too, were rarely men of blood and iron but usually bookish and learned themselves, and generally quite interested in what these heretics thought and why. Emmanuel LeRoy Ladurie's book is good on this.
Generally, in front of the papal inquisition, you had quite a good chance of being able to talk your way out of trouble- provided you were actually innocent (by their standards), they would likely let you go; they were genuinely concerned about guilt and innocence in a way we don't see again until at least the rise of Scotland Yard.
...And the Spanish Inquisition was established, largely to clear out the crypto- Islamics from Spain in the wake of the Reconquista, in the belief that the Papal Inquisition had been far too soft, these were harder times and much harder measures were necessary and permissible.
The operations of the papal inquisition were hardly peaceful, considering they were often accompanied and preceded by military force, but the Spanish inquisition proceeded on the assumption of malice, chasing hidden traitors rather than lost sheep, and things did become much more brutal on the ground.
The Jesuits are established after- and make no sense without- the Reformation; they were both a reform movement within the Church and a counterattack against Protestantism, and some of the variations on Christianity they gave rise to, with missionary activity being taken by the locals in their own ways, are so fascinatingly strange they would have made a papal inquisitor's brain curl up. (The church has frequently regarded the Jesuits as too clever for their own good.)
Anyway, without the Spanish Inquisition, well, Spain would not have been entirely cleared of Moriscos and Jews, and the Spanish people would not have been taught to fear their own government- not quite so much, anyway.