I'll talk about what a bishop is in the Episcopal Church. A bishop is the fourth in the four orders of ministry. The first three are the laity (regular folks), the deacons (more on them later), the priests, and the bishops.
I like to think of a bishop as the pastor for an unusual church. The bishop's church is the diocese. A diocese is a collection of churches, each church is a member of that diocese. Each church is an individual that is part and parcel of that larger community.
The current bishop of Atlanta is Neil Alexander. I meet with him on Jan. 17th to finalize the end of my aspirant stage and start the postulancy stage and talk seminary. A bishop has his job for nine years.
I guess you could say that the job title of a bishop is to maintain unity in the diocese. To do this, the bishop travels alot. I once heard the bishop say, "To be bishop you must love God and like to drive.
Above the diocesan bishops is the Presiding Bishop. The presiding Bishop is Katherine Jefferts-Schori, she was elected (by all the other bishops) in 2006. The Episcopal church does not have arch-bishops. Katherine is the bishop for the episcopal church in the U.S.
Bishops ultimately attempt to hold the union, or communion, of the church. This communion has been strained over the past few years, especially between the Episcopal Church in the U.S. and the Anglican Church. It's gotten so bad that some churches and even a few dioceses(local collection of churches) have sought out other bishops to give them oversight. There is a serious canon violation (canon's are church laws)when another bishop comes into another country to give oversight. More on this later.