Well, the origins of the Illuminati are Bavaria, and their mission was separating the church from the govt, and limiting the power of federal government to prevent overreach. Obviously the Catholic Church viewed them as enemies for that, as it was the Church who crowned monarchs of the time, and they wielded immense political power in that time.

I dont really know what it is now though; if it's really the name for the globalist cabal, or if that's a shifting of the blame by the cabal to discredit an organization that would be diametrically opposed to it, under the founding principles.