Once completed, the Order will recognize you as a member of the Psijic Order, welcome to research, train, and stay on the island of Artaeum. You will meet Sargenius, who will then begin the meticulous initiation tests, which will include getting Soul Sand (finely ground soul gems) from a Dremora, killing a Daedroth using only the storm spells, disenchanting daggers without losing the dagger, enchanting without soul gems, and other seemingly impossible things.
When cast, it will teleport you to Artaeum, costing at least 750 magicka to cast. They'll contact you and you'll be given a mysterious spell. I think that in order for them to contact you again, you must be at level 80 or higher and/or have gone legendary in five of the following skills: Okay, so the Psijic Order is an incredibly powerful league of mages that have magic that can send entire islands into another plane of existence or even stop time and teleport across Tamriel.