The Bishops of the Church are given to us by Christ to act as our shepherds. Bishops are in charge of a "Diocese" which is (usually) a geographical area made of local communities (parishes) of the faithful. The Bishop is the superior of all the priests (except those belonging to religious orders) in his dioceses, as well as the local spiritual leader of all the faithful of his diocese.
In an Archdiocese, the Bishop will usually be accorded the honourable rank of Archbishop. The usual way to address an Archbishop in Commonwealth countries is as "Your Grace" ("Your Excellency" is more usual in North America) and to address a Bishop is as "My Lord" ("Monseigneur").
In the modern Church, however, not all bishops occupy pastoral positions as the heads of dioceses and several dioceses are so large as to require more than one bishop rather than divide the diocese into smaller dioceses. Thus many Bishops are Papal Diplomats, and many Dioceses have one or more "Auxiliary Bishops" assisting the Bishop.
Bishops are always chosen and appointed by the Pope (although in some of the Eastern rites, Bishops are actually appointed at a synod of Bishops and the choice ratified by the Pope). The Process is usually that the Papal Nuncio for the relevant region will provide the names of three candidates to the Congregation for Bishops in the Vatican who advise the Pope. These three candidates are usually chosen in consultation with the outgoing Bishop (unless, of course, the see is vacant because of the death of the incumbent bishop).
All Bishops (except the Pope, Bishop of Rome) are required by Canon Law to tender their resignation if sickness or other grave reasons make them incapable of carrying on their role, or when they reach the age of 75.
All Bishops are also required to submit a quinquennial report to the Pope (ie, every five years) reporting on their diocese and any problems that may have arisen in their diocese or difficulties the faithful are facing. At about the time that this quinquennial report is required, the Bishops of the region make their visit ad limina Apostolorum where they travel to Rome to pray before the Tomb of St Peter and to meet individually with the Holy Father to ensure he is kept aware of the state of the Church throughout the world.
Most countries or regions have established Episcopal Conferences or Bishops' Conferences. These are forums for the bishops of a region to come together to discuss issues of mutual concern. It should be remembered, however, that it is not the Episcopal Conference that exercises the teaching and pastoral duties, but the individual bishop in his individual diocese.
Because all Bishops must have a diocese under their care, those Bishops who are not in charge of a "real" diocese of their own are given "Titular Episcopal Sees", that is, dioceses of which they are the titular head. These Dioceses are dioceses which once existed in the countries surrounding the Mediterranean but are now defunct.
The media tried to make a big thing about Bishop Jacques Gaillot's being removed from his diocese in France a few years ago, because in the process he was made Titular Bishop of Partenia, a defunct diocese in the Sub-Sahara. This was supposed somehow to be especially bad treatment of him by the Vatican. What they didn't make clear was that every bishop in an administrative, diplomatic or auxiliary post around the world has a titular diocese somewhere in North Africa, the Holy Land or Mediterranean Europe!
Apart from Auxiliary Bishops assisting a Bishop or Archbishop in the care of a diocese, there can also be "Coadjutor Bishops". They are appointed by the Pope to assist a Bishop or Archbishop in the care of a diocese, however, there are important differences:
Coadjutor Bishops are only rarely appointed and only for serious reasons, usually where the incumbent Bishop is quite ill or Rome has grave concerns about the incumbent Bishop's care for the faithful entrusted to his care.