An Archbishop has a limited right to supervise and chastise his suffragan bishops, and their priests. One has the right to appeal to one’s bishop, then one’s archbishop, then to Rome, if need be.
In practical terms, it means little, besides commemoration in the liturgy, for the average Catholic.
Cardinal is a title assigned to those granted status as Electors of the Pope. It has nothing to do with the local diocese.
It is not restricted to priests (though almost always bishops, historically, priests and deacons have been made Cardinals with a vote), and they only vote if below a certain age (ISTR it being 75). Certain laymen are also granted the title Cardinal, usually for a life-long service as a layman-Theologian or layman-Cannon-lawyer, and never before being older than voting age (so they do not vote, and thus do not elect the Pope), though it is theoretically possible that some could be.
The Eastern Rite Catholic Patriarchs are usually granted cardinal title from the order of Cardinal-Bishops, when such a titular see becomes open. (See the CCEO)
A diocese is a defined geographical area headed by a bishop.
An Archdiocese is the diocese headed by an Archbishop.
An Ecclesiastical Province is an Archdiocese and its related suffragan Dioceses, and is headed by the Archbishop.
A Patriarchate is either:
A “super province” headed by a particular special Archbishop, whose title is Patriarch, and to whom there are suffragan Archdioceses/Provinces, (Eastern Rites) or
A title of special honor for certain historical dioceses’ Bishops (Western Rites).
The term “Ordinary” refers to a Diocese’s Bishop-in-charge; so for a diocese, it’s the Bishop, for the Archdiocese, it’s the Archbishop. Note that Eastern Patriarchs have “ordinary jurisdiction” over all their suffragan bishops; western ones do not. The Pope has “ordinary jurisdiction” over the whole of the Catholic Church.
Certain non-bishops also are sad to have Ordinary Jurisdiction in very limited cases, but are not themselves Ordinaries.
An Ordinary is allowed to ordain priests & deacons, set the rules for posture of the faithful during the liturgy (within their see), issue and relieve bans, supervise courts of canon law, and certain other duties.
Auxillary Bishops are bishops elevated to assist another bishop. They are not “ordinaries.” They hold title to a diocese that doesn’t exist anymore As a functioning diocese.
The term “Particular Church” is a reference to an Ordinary, his diocese or archdiocese, and all the faithful of that diocese.
With the Eastern Rites, there are some variations on this, but in general, they work the same way.
Hisotrically, it also took 2 archbishops or 4 bishops to enthrone a king.