When “ranking” hierarchs, it is best to simply leave cardinals aside as their own parallel track in the hierarchy. Here’s my run-down.
Bishop: All bishops possess the same level of Holy Orders, be they bishop, archbishop, or cardinal. Those who are titled bishops might either be the ordinary (man-in-charge) or a diocese or the auxiliary of a diocese. (AFAIK it is not normal to ordain men as “simple” bishops outside of those two cases). They possess the fulness of the priesthood, and their authority is confined to their own diocesan territory.
Archbishop: Archbishops are bishops who hold special dignity within the Church. There are three main classes of archbishops. The first is metropolitans. Metropolitans are archbishops who are ordinaries of the head see within an ecclesiastical province. For instance, Indianapolis is the metropolitan see of Indiana, and Abp. Buechlein is thus the head bishop of Indiana. In modern times metropolitan archbishops have little concrete authority outside of their own territories, whereas they used to have more sway. Under the current
CIC, metropolitans duties and privileges in their subordinate dioceses are:
Can. 436 §1. In the suffragan dioceses, a metropolitan is competent:
1/ to exercise vigilance so that the faith and ecclesiastical discipline are observed carefully and to inform the Roman Pontiff of abuses, if there are any;
2/ to conduct a canonical visitation for a cause previously approved by the Apostolic See if a suffragan has neglected it;
3/ to designate a diocesan administrator according to the norm of cann.
⇒ 421, §2, and
⇒ 425, §3.
§2. Where circumstances demand it, the Apostolic See can endow a metropolitan with special functions and power to be determined in particular law.
§3. The metropolitan has no other power of governance in the suffragan dioceses. He can perform sacred functions, however, as if he were a bishop in his own diocese in all churches, but he is first to inform the diocesan bishop if the church is the cathedral.
A second class of archbishops are those who head “important” or large dioceses that have no suffragan (subordinate) dioceses attached to them. In the United States, the archdioceses of Washington, D.C. and for the Military Services are two prime examples. These archbishops claim precedence of honor over other bishops, but have no responsibilities or privileges outside their territory as do metropolitans.
A third class of archbishops are curial officials, in particular the heads of dicasteries and apostolic nuncios. Once again, these claim precedence of honor, and they furthermore are not ordinaries of a dioceses. I believe they are ordained to the episcopate and given the honor of archbishop because their duties will require them to “give orders” to other bishops in the world, and thus it was felt that propriety demanded they actually be “superior” to those bishops in some respect.
You can see, then, that because not all archbishops have dioceses and not all of those actually have authority over other bishops, even here its not fully correct to simplify things into a “ranking” of bishop, archbishop, pope. If one is confining oneself to honor and precedence, this would be accurate, but if one is trying to map the hierarchy of jurisdictional power, it would run bishop, metropolitan (archbishop), pope, with archbishop type 2 ranking with bishops and type 3 sort of floating to the side (they exercise
papal authority over bishops and archbishops but none in their own right).
Cardinals, now, further complicate things. According to modern law, all cardinals will be bishops (and, in practice, archbishops) with a few exceptional priests. But in theory, cardinals need not even be deacons. It used to be considered normal for laymen to be named cardinals - including one of the most famous cardinals of the sixteenth century, Cdl. Pole, who was cousin to Henry VIII and sometime papal legate to Trent (he did, eventually, receive holy orders but for many years he was a cardinal while not yet even a deacon). The cardinalate is an honorary designation that makes one a member of the pope’s “senate,” his group of counselors, and a papal elector. It also carries precedence of honor over all other non-cardinals (except, perhaps, patriarchs of Eastern churches, though I can’t remember for sure), along with certain privileges of exercising what orders one has anywhere in the world.
Cardinals are typically named for three reasons. The first is that they are the metropolitans of significant archdioceses. Certain archdioceses, like New York, Westminster, Paris, Milan, etc., are considered “automatic” cardinal’s positions, although such fortunes do shift as ecclesiastical demography does as well. The second is that they are high-ranking and/or long-serving curial officials. Here the prefectures of the CDF, CDW, and Apostolic Signatura are the sorts of things that bring an “automatic” red hat. The third group are distinguished theologians, typically only in their advanced years and thus after not just a distinguished but also long service in that capacity. Although these men, typically not bishops, are required to be ordained bishops upon their ascendancy to the cardinalate, the law provides that the pope might allow them to remain in the presbyterate if they so request - the best known example is the recently deceased Cdl. Dulles. These priest-cardinals show perfectly why it can be confusing to try to rank cardinals with bishops and archbishops, because Cdl. Dulles held precedence over the bishops of the world but never exercised authority over anyone.
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Perhaps reluctance to invest such authority in such a young country