What in the name of Eris are the Illuminati?

Briefly, the background of the Bavarian Illuminati puzzle is this.

On May 1st, 1776, in Bavaria, Dr Adam Weishaupt, a professor of Canon Law at Ingolstadt University and a former Jesuit, formed a secret society called the Order of the Illuminati within the existing Masonic lodges of Germany. Since Masonry itself is a secret society, the Illuminati was a secret society within a secret society, a mystery inside a mystery, so to say. In 1785 the Illuminati was suppressed by the Bavarian government for alledgedly plotting to overthrow all the kings in Europe and the Pope to boot. This much is generally agreed upon by all historians. Everything else is a matter of heated, and sometimes fetid, contoversy. 

It has been claimed that Dr Weishaupt was an athiest, a Cabalistic magician, a rationalist, a mystic; a democrat, a socialist, an anarchist, a facist; a Machiavellian amoralist, an alchemist , a totalitarian and an "enthusiastic philanthropist". (The last was the verdict of Thomas Jefferson, by the way.) The Illuminati have also been credited with managing the French and American revolutions behind the scenes, taking over the world, being the brains behind Communism, continuing underground up to the 1970s, secretly worshipping the Devil, and mopery with intent to gawk. Some claim that Weishaupt didn`t even invent the Illuminati, but only revived it. The Order of the Illuminati has been traced back to the Knights Templar, to the Greek and Gnostic initiatory cults, to Egypt, even to Atlantis. The one safe generalization one can make is that Weishaupt`s intent to maintain secrecy has worked; no two students of Illuminology have ever agreed totally about what the "inner secret" or purpose of the order actually was (or is...). There is endless room for spooky speculation, and for pedantic paranoia, once one really gets into the literature of the subject; and ther has been a wave of sensational "exposés" of the Illuminati every generation since 1776. If you were to believe all this sensational literature, the damned Bavarian conspirators were responsible for everything wrong with the world, including the energy crises and the fact that you can`t even get a plumber on weekends.

For instance, the first explosion of anti-Illuminati hysteria in this country, in the 1790s, was stirred up by fanatic Federalists and centred on the charge that Thomas Jefferson and the Democratic Republican party were pawns of the European Illuminati. The second major cluster of excited exposés came in the 1840s, and was circulated by the Anti-Masonic party, who believed that the Illuminati still controlled the Masons and had infultrated our government at all levels. In both of these instances, the Illuminati were portrayed as radical democrats or outright anarchists in the tradition of the ultra-left wing of the French Revolution. Current anti-illuminati theory, occasionally interacting with this political conspiracy literature, portrays the Illuminati as Nazis, black magicians, astral mind-fuckers and Satanists.