SCHOOL is a group in Academy City which develops a rivalry with ITEM in the Battle Royale Arc of the series. It is commanded by the #2 ranked Level 5 Kakine Teitoku. The original purpose of the organization is not known, and by the events of volume 15 they are apparently bent on "negotiating directly with Aleister Crowley", as inferred from a conversation between Kakine and the Girl in the Dress . As Accelerator stated, "It's an organisation hidden in Academy City's underworld" and is similar to GROUP. So it is likely the group takes orders from people higher up in Academy City. The group was supported by a subordinate organisation that assists SCHOOL in its operations.
It is currently unknown why the group is called SCHOOL, but it presumably has something to do with school (similar to how ITEM has members that use items in combat).
At the present, the group is probably considered broken up after the defeats of many of the members (namely Kakine Teitoku, Suzana Chimitsu and the Boy with Goggles) in Toaru Majutsu no Index Light Novel Volume 15. The Boy with Goggles was killed by ITEM, Sunazara Chimitsu by being injured and Teitoku by combat with Accelerator.