Initiation was a process by which a new member would be inducted into an organisation.
According to the Tenth Doctor, all children on Gallifrey were taken from their families at the age of eight and, before they went to the Academy, they were taken for initiation; they were made to stand in front of the Untempered Schism, staring at "the raw power of Time and Space". Although some would be inspired, others would run away or go insane. (TV: The Sound of Drums)
Faction Paradox initiations, meanwhile, involved a more practical demonstration of the skill and inventiveness the prospective new Cousin would bring to the Family; one was expected to craft a particularly interesting paradox. (PROSE: Daring Initiation)