What is the difference between a cardinal-priest and a cardinal-bishop?
Cardinals are Bishops. Bishops are Priests.
Most Priests are ordained Deacons before they are ordained Priests. Most Bishops are ordained Priests before they are ordained Bishops. Most Cardinals are ordained Bishops before they are ordained Cardinals. And most Popes are ordained Cardinals before they are chose to be Pope (though technically any Baptized male Catholic can be elected Pope).
It is not a requirement that Cardinals must be a Bishop, but one would need to receive a papal dispensation in order to become a Cardinal without first having been ordained a Bishop.
Ahhh so a cardinal-priest skipped being a bishop?
But then how can a Cardinal-priest also become an Archbishop?
How is an Archbishop different than a Bishop?
I don’t know whether cardinal-priests can become archbishops without becoming bishops first. My understanding of an archdiocese is that, beacause of its size, it has more than one bishop, and the top bishop is the archbishop. For example, I live in the Philadelphia Archdiocese, which has an archbishop, and, I think, four or five auxiliary bishops.
An Archbishop oversees fellow Bishops in a particular Region (made up of several Dioceses).
For example, in Ohio, there are six Dioceses and the Archbishop is the Bishop of Cincinnati, Archbishop Dennis M. Schnurr.
Is there a particular Cardinal that you are referencing that sparked your question? Maybe we can do a deep-dive into him specifically.
This page shows a list of all current Cardinals
It says there’s 3 types: Cardinal-bishop, Cardinal-priest, Cardinal-deacon