The Inquisitors were originally intended to offer a parallel, but non-Force based, capability to the Jedi Hunters. While the Hunters went after trained Force users, the Inquisitius targeted thier families, potential students and thier support organizations (such as the Freedom's Sons [aka Freedom Warriors]), while generating and fostering anti-Jedi sentiment among the populace. With the eventual completion of the main phase of the Purge, the Jedi Hunters have by and large been shut down, with at least one of former Hunter becoming part of the Inquisitius.
Today, the Inquisitius has three major missions. The first is to support Imperial Intelligence and the ISB in the questioning of particularly resistant suspects, or those believed to have Force abilities. In this capacity, the Inquisitors have become infamous among Rebel personnel.
They also continue thier mission of discrediting the Force and and denying it's existence. On certain, primitive worlds were similar beliefs are common, Inquisitors wage a not so subtle war, both hot and cold, against the local theocrats. (An example of this is Barab, where local legends speak of Jedi intervention to prevent a war among the aboriginal primitives. The Empire has repeatedly stated that the Jedi were actually searching for slaves.) They have gone so far as to attack religious and learning centers, killing all those who would pose a threat to the New Order, and blame the Rebellion, since the Jedi are little more than a memory now. (During the Purge, they destroyed the Jedi facility on Durok-Tian VII, and left the body of a known Teepo Paladin in the rubble.)
Finally, the Emperor is obsessed with collecting primitive (and not so primitive artifacts) that are strongly touched by the Force (such as Jedi's ships and weapons, texts, mystical items, and the like) or are based on technology in excess of the Imperial standard (such as the thirty agents assigned to the Gree Enclave). The archaeological teams are little more than grave robbing incompetents, and are still trying to live down the scandal with the Voorlach "Temple" on Krykis IV.
No matter what the current assignment of an Inquisitor, they are charged with powers of arrest, summery trial and sentencing as they see fit. The Inquisitius answers only to the Emperor and small handful of his advisors, including Lord Vader. They are believed to be responsible for hundreds of thousands of innocent deaths, with another ten sentenced to such places as Daspyre, Kessel, Dathomir and Sevarcos for every one sentenced to die.
An Inquisitor is roughly equivalent of an army Major General. They are assigned to a certain location and mission (such as the observation and retrial team in Gree Space). Each individual agent is able to call on up to a platoon of Stormtroopers, Dungeoneers or COMPForce troops for his personal guard, and are known to occasionally hire mercenaries or bounty hunters. Rebel Intelligence estimates that there is perhaps as many as 500 Inquisitors.
The next step up is the rank of High Inquisitor. These 12 evil beings are assigned an area to patrol, referred to as a "hunting reserve" by Rebel command. These men can requisition just about any amount of conventional forces they wish for thier personal use, including a small number of special operations personnel (Storm Commandos, Amry Special Missions, SpecNav, etc.). They are always assigned a small detachment of Imperial ("Coruscant") Guards for thier entourage, and many have thier own "trouble shooters" or bodyguards. Finally, they always have an Inquisitor or two with them. They can draw thier own transportation, within limits. (These limits seem to vary from individual to individual. For example, Treymaine has his own ISD II, while Slosin has supplied his own Trenchant-class ship.)
Regardless of what dealing you have with the Inquisitius, be warned: you will most likely not enjoy it.