Not to worry; a bishop is just a bishop.
A bishop who is assigned to run a diocese (“bishop” means “overseer”), he is a diocesan bishop, known as the Bishop of Wherever. He will automatically have various rights and duties by virtue of having become a diocesan bishop (e.g., he has to care for the priests of that diocese, and he can appoint pastors to parishes within that diocese, etc.).
If he happens to be assigned to an archdiocese instead of a diocese (that difference being one of size, importance, or history), then he’s an archbishop, known as the Archbishop of Greater Wherever.
If he resigns/retires (and the resignation/retirement is accepted by the Pope), then he becomes Bishop Emeritus of Wherever.
If he’s assigned to another bishop’s diocese to help out, he’s an auxiliary bishop. That’s sort of like an assistant bishop; he has all the powers of the bishop, but he does not have the right of succession to that diocese. He’s simply the Auxiliary Bishop of Wherever.
If he’s assigned to another bishop’s diocese to help out and it’s understood that he will be the next bishop of that diocese when the current one leaves, then he’s the coadjutor bishop of that diocese. Then everyone knows he’s the next Bishop of Wherever.
Coadjutor bishops and auxiliary bishops are described in the Code of Canon Law and have certain rights and duties associated with their positions.
A bishop who is not assigned to care for a particular diocese is called a titular bishop: he has the title and powers of a bishop but not a diocese to go with them (and therefore lacks the powers associated with being a diocesan bishop: he can’t appoint a pastor of a parish, since he doesn’t have any parishes).