The first, known as the Medieval Inquisition, was established in 1184? in response to the Catharist? heresy in southern France and faded in power with the successful suppression of that group.
The infamous Spanish Inquisition was not an ecclesiastical operation at all. It was established in 1481 by Ferdinand and Isabella to investigate and punish the Jews and Moors? who had publically converted to Christianity but had privately continued to practice their prior religion and were thus, by definition, heretics. Jews or Muslims who did not become Christians were never subjected to the powers of the Inquisition.
The Roman Inquisition, begun in 1542, was quite distinct from either of these. It was the least active and most benign of the three variations.
further reading: Edward M. Peters, Inquisition. (University of California Press, 1989). ISBN 0520066308 (amazon.com, search)
Henry Kamen, The Spanish Inquisition: A Historical Revision. (Yale University Press, 1999). ISBN 0300078803 (amazon.com, search)