|We must've been through hundreds of books already and we can't find him anywhere —|
The Deputy Headmaster or Deputy Headmistress is a title given to the faculty member at Hogwarts School of Witchcraft and Wizardry that have been given special responsibility in regard to assisting the Headmaster or Headmistress of the day with the administrative running of the school in addition to their duties as teachers. Among the tasks entrusted to them are the annual visit to a small locked tower in the castle housing the Quill of Acceptance and the Book of Admittance in order to check the names of young witches and wizards to whom they would have to send a letter of acceptance along with a list of school supplies approved and signed by the Chief Attendant of Witchcraft Provisions. The Deputy Head also presumably were the ones entrusted the task of representing the school in meetings with the parents of Muggle-born witches and wizards to inform of the existence and introduce them to the wizarding community of Great Britain.
In the event of a headmaster's incapacitation, the deputy headmaster serves as temporary headmaster until the Board of Governors can appoint a new, permanent one, though it is likely that the Deputy Head will more often than not be a preferable choice over any outsider due to their familiarity with the school's affairs and proven ability to attend to administrative and academic matters alike, though it is not necessarily always the case.