Cathedral

A Cathedral is the seat of a bishop. Every bishop should have one.
Beyond that, it is an ecclesiastical building, of the Christian persuasion, more important than a church, which is itself one up from a chapel. That said, functionally, one would expect that a church would have at least a priest appointed to serve it, and that a chapel would have some degree of appointed guardian and minster, whether a priest, a deacon, a hermit, or just a spare saint.

Technically, St.Peter's in the Vatican City is the cathedral of the Bishop of Rome.