Every Cardinal and Archbishop is a Bishop.
A Bishop’s job is to be the ecclesiastical authority and chief pastor for a diocese. There are a couple of things that only he can do: confirm and ordain, and a number of things left to his proper discretion, ranging from approving interfaith marriages in Catholicism to consecrating chrism.
Historically, the dioceses over which bishops preside are grouped together into provinces, AKA archdioceses, each with an archbishop. The archbishop has some authority over the bishops in his province. (There are two slightly contradictory uses of the term archdiocese: it can be synonymous with province, meaning the half dozen dioceses over which an archbishop has authority; or it can mean the diocese of which the archbishop is the bishop – since he is an archbishop, it’s an archdiocese. Anglicans always use it the first way, as Catholicism historically did; contemporary Catholic usage tends to mean the second way.) [Aside: Eastern Rite Catholics and Eastern Orthodox have Metropolitans which are exactly the same thing as Archbishops; it’s merely a different term used in churches of the Eastern tradition.]
A Cardinal is a Bishop, nearly always an Archbishop, who has been named as one of the “Princes of the Church” in Catholicism. He gets an honorary title relative to one of the churches in Rome, the right to wear the red hat and clerical garb, and a certain amount of authority relative to the church as a worldwide whole, most especially the right to vote for a new Pope.
They’re all doing the job of a bishop, but some have additional duties and an additional honorific in token of that.
There are a couple of odd and confusing details to how Catholicism does things, though:
There’s a longstanding tradition that ties a bishop to a city over which he has episcopal authority, so the rule is one bishop per city. However, this runs up against the fact that some dioceses really need two or more bishops. So if a suffragan bishop is named to assist the Bishop of, say, Kansas City, his job and his office are there in Kansas City, but to preserve the one bishop/one city distinction, he is named Titular Bishop of Ancient-City-in-the-Moslem-World-that-used-to-be-a-see-city. The incumbent Bishop of Ogdensburg, NY, used to be the suffragan of the Bishop of Buffalo, and was at that time Titular Bishop of Hippo Regius, in succession to St. Augustine.
Then there’s the Archbishop ad personam, which simply means that somebody “deserves” to be elevated to Archbishop in the Pope’s estimation, but there’s no Archdiocese open that needs him. So he remains bishop of one diocese, but is officially designated as archbishop, by way of promotion.
The Pope is, among other things, Bishop of Rome; he’s head of that (arch-)diocese, though 99% of the work of episcopating over that diocese is delegated to other bishops. (Before his health failed, John Paul used to, every so often, go out and hear confessions, celebrate Mass, and/or do confirmations in Rome, giving real meaning to that tradition, another of the things for which he was beloved by Catholics.)
It’s not customary but not improper to refer to an Archbishop or a Cardinal as Bishop X; he remains a bishop, just one with additional dignities.
A Cardinal in and of himself, by the way, has no authority over Bishops or Archbishops except the ones in his own archdiocese. But he may have a churchwide job that gives him some authority over all Catholics, as in being one of the people whom the Pope depends on to review and approve annulments, the ordination of converted clergy, etc.
Final oddball point. The term Patriarch may come up. There were originally five Patriarchs, in Alexandria, Antioch, Jerusalem, Rome, and Ephesus. The one at Ephesus was early transferred to Constantinople. One of the Pope’s other titles is Patriarch of the West. However, the number of Patriarchates grew over the years: several of the national churches of Orthodoxy have Patriarchs, including notably the Russian Orthodox Patriarch of Moscow. And as Eastern Rites were created or returned to Catholicism, Patriarchs were recognized, so that Alexandria, for example, has three Patriarchs, one for the Coptic Orthodox, one for the Eastern Orthodox, and one for the Eastern Rite Catholics. In addition there are a very few Archbishops who have that dignity in the Western Church: Venice and I think Milan have Patriarchs, and there may be another one.