A bishop is a priest who carries the fullness of the sacrament of Orders (Decree on the Bishops’ Pastoral Office in the Church, n. 15). He usually heads a diocese and is called an “ordinary” because he promotes order, and is in charge. An auxiliary bishop is one who assists the diocesan bishop. A coadjutor bishop is appointed to assist a diocesan bishop and has ‘the right of succession’ when the diocesan bishop retires or dies. In the diocese of Lafayette-in Indiana, Bishop John Joseph (later Cardinal) Carberry was the coadjutor to Bishop John George Bennett until 1957.
A man becomes an archbishop, in most cases, because he has been named as the head of an archdiocese. Some archbishops have the title because they oversee important administrative functions or they receive the title as an honor.
The title cardinal is given to members of the College of Cardinals. The appointment is made by the pope, and his most notable role will be to serve as a papal elector. It is a lifetime title, but the ability to vote for a successor pope ceases at a cardinal’s 80th birthday. With the most recent appointments, there will be about 125 voting age cardinals worldwide. Most of the cardinals are archbishops, but occasionally a pope will bestow the honor on a scholar or an outstanding priest.
As a body, the cardinals provide the pope some assistance in the administration of the church. Their title brings some extra responsibilities in their own countries and within the worldwide church.
The word cardinal seems to have roots in the Latin word cardo, which means hinge. One might surmise that the good of the church and the succession of popes hinges on the members of this College.
Different resources will offer more detail about these titles. The New Catholic Encyclopedia (2002) and The Encyclopedia of Catholicism (1995). See also the website www.catholiceducation.org