Dear Lance,
The pope’s authority covers the whole Church, including bishops and priests. However, practically speaking, each priest is accountable on a daily basis to his bishop (and religious superior if he belongs to a religious order). There is no way that the pope could over-see the behavior and ministry of each priest in the Church. Each local ordinary (local bishop), though accountable to the pope, has a certain freedom to minister in a way that he sees is most effective in the face of the unique needs of his diocese. His priests are expected to co-operate with him in this. So they promise to obey him. It is understood that such obedience is in concert with the bishop’s obedience to the Holy Father.
Fr. Vincent Serpa, O.P.