This is about Sabbat Archbishops. For information on Cult Archbishops, See Cult of the Emperor.
Archbishop is a title used in the Sabbat hierarchy, a very roughly equivalent to the Camarilla's Prince. They function as upper middle management in the Sabbat, above Bishops but below Cardinals. Archbishops are assigned an archdiocese to oversee, usually either a single densely-populated city, or a larger but less populated area (for example, Chicago vs. downstate Illinois) within their Cardinal's region. Their orders are carried out and enforced by the Bishop of each diocese within the archdiocese.
Cardinals can appoint any vampire they want to an Archbishop position, although qualified candidates usually rise through the ranks, serving lengthy tenures as Priests and then Bishops. Some Cardinals accept bribes to bestow the title on someone, although vampires who acquire the position this way often mysteriously disappear.
In addition to administering their archdiocese, Archbishops also appoint Bishops, initiate, promote, and demote Sabbat members of lower rank, and advise their Cardinal.