The Organization is a crime syndicate founded in Japan by Nejirebane and her adopted son Ushijima Masashi some time after World War II. They control the country from the shadows and seek to weed out those they find worthless, to the extent of setting off major catastrophes to deplete large amounts of the population.
Their enforcers are killer-for-hires who have some sort of skill, supernatural or not, that is related to arthropods. They have several children at their disposal on orphanages that are trained in the ways of assassination by Bugs such as Kumo and Kabutomushi.
Over the course of Arachnid, Nejirebane tasks Guntaiari with developing a sex-based zombie apocalypse to kill half of Japan's population from starvation and allow the Organization to establish their ideal society over the ruins. At the same time, they begin a competition called "Arachnid Hunt" among the assassins with the goal of killing Kumo's pupil, a deadly schoolgirl named Fujii Alice. The real purpose of the hunt, however, was to spur Alice's growth and force her to become the final Suzumebachi companion to Nejirebane.
After Nejirebane is killed by Kabutomushi at the end of the Arachnid Hunt, the high-ranked members begin seeking out a replacement, with the implication they expect Alice to fill the role one way or the other.