The Role of Catholic Bishops and Archbishops
In the Catholic Church, archbishops and bishops rank below cardinals. Becoming a bishop is the third and fullest level of the Sacrament of Holy Orders. The first level is the ordination of a deacon, the second is the ordination of a priest, and the third is the ordination of a bishop. A bishop who moves to the level of cardinal isn’t ordained, but handpicked by the pope, who also appoints bishops.
A bishop oversees a diocese, which is a collection of local parishes; and an archbishop administers an archdiocese, which is just a really large diocese. (Denver, Hartford, Omaha, Miami, Newark, St. Louis, and San Francisco are examples of archdioceses.)
You can think of a local parish like a town or city, and the local pastor — a priest — is like the mayor. The diocese is like a state or province, and the bishop is like the governor. An archdiocese is like a very populous state — California or Texas, perhaps. (The pope is like the prime minister, governing the entire nation, except that in this case, the nation is really the entire world.)
Each bishop must make a visit to the Holy Father every five years and give a report on his particular diocese. The bishops within an entire country or nation get together at least once a year in a gathering known as an episcopal conference.
The rest of the time, the bishop goes around the diocese performing the sacraments of Confirmation and Holy Orders. (Only bishops have the authority to administer the Sacrament of Holy Orders whereby men are ordained deacons, priests, or bishops.) Bishops make visits to the parishes and chair numerous meetings with their staff.
Each individual bishop has his own authority to run the diocese. He’s not an ambassador of the pope but governs the local diocese as an authentic successor of the apostles, just as the pope governs the universal Church as the successor of St. Peter.