The Persona Compendium is a service offered by Margaret in the Velvet Room allowing you to purchase personas that have been registered once you unlock them through fusions or collecting their card during one of the random 'Shuffle Times' that occur after a battle.
The cost of the persona depends on its level and you cannot buy a persona that is in your inventory. It needs to be a persona you used in a fusion or one you discarded to make room for another.