In what ways is the term "priesthood" used in Catholic tradition?

The Council of Trent, session 23 on Holy Orders, can. 1 calls the ministerial priesthood (≠ universal/common priesthood or priesthood of the faithful)

in the New Testament a visible and external priesthood (sacerdotium visibile et externum)

Pope Pius XII, in Mediator Dei §83, calls it a

hierarchical priesthood (hierarchicum sacerdotium)

In what ways is the term “priesthood” used in Catholic tradition?

Traditionally, there are only two ways that term priesthood is used within the Church as well as one in the Ancient Covenant of the Jewish people of the Old Testament, known as the Aaronic priesthood.

The priesthood of the New Covenant is distinctly different from the priesthood of the Old Testament.

Yet other priesthood definitions are on the horizon.

The Jewish priesthood

In the age of the Patriarchs the offering of sacrifices was the function of the father or head of the family (cf. Genesis 8:20; 12:7, etc.; Job 1:5). But, even before Moses, there were also regular priests, who were not fathers of family (cf. Exodus 19:22 sqq.). Hummelauer's hypothesis "Das vormosaische Priestertum in Israel", Freiburg, 1899) that this pre-Mosaic priesthood was established by God Himself and made hereditary in the family of Manasses, but was subsequently abolished in punishment of the worship of the golden calf (cf. Exodus 32:26 sqq.), can hardly be scientifically established (cf. Rev. bibl. internat., 1899, pp. 470 sqq.). In the Mosaic priesthood we must distinguish: priests, Levites, and high-priest.

The Christian priesthood

In the New Testament bishops and priests are, according to Catholic teaching, the sole bearers of the priesthood, the former enjoying the fullness of the priesthood (summus sacerdos s. primi ordinis), while the presbyters are simple priests (simplex sacerdos s. secundi ordinis). The deacon, on the other hand, is a mere attendant of the priest, with no priestly powers. Omitting all special treatment of the bishop and the deacon, we here confine our attention primarily to the presbyterate, since the term "priest" without qualification is now taken to signify the presbyter. - Priesthood (Catholic Encyclopaedia)

The Church also speaks of the Royal Priesthood of the Faithful.

The share in the high priesthood of Christ received by everyone at baptism and strengthened by confirmation and the Eucharist. Essential to this priesthood is the right to receive the other sacraments, of participating in the Church's liturgy, and of being united with Christ the eternal priest as he offers himself, with the members of his Mystical Body, to the heavenly Father in the Eucharistic Sacrifice. - Priesthood of the Faithful

The terms simply do not end here. As the Church moves forward, new ways of expressing the term may be on the horizon.

The priesthood of Mary is seen as being part of the universal priesthood of the faithful, but at the same time needs a clearer understand which may come to light in future years.

After the council, the priesthood of Mary is still spoken of, but it is no longer linked to the ministerial priesthood nor to the supreme priesthood of Christ, but rather to the universal priesthood of the faithful. As figure and first fruits of the Church, she possessed in a personal way that "royal priesthood" (1 Peter 2:9), that all the baptized possess in a collective way.

What can we retain of this long tradition that associates Mary to the priest and what meaning can we give to the "closeness " between Mary and the priest affirmed by John Paul II? It seems to me that what remains is the analogy or the correspondence between the different dimensions of the mystery of salvation. What Mary was once and for all times on the plane of historical reality, the priest is ever anew on the plane of the sacramental reality. - Mary, Mother and Model of the Priest - Cardinal Raniero Cantalamessa, OFMCap, Pontifical Household Preacher

What terms the Church will ultimately use to express Mary’s unique role within the priesthood of the faithful remains to be seen!

The Blessed Virgin Mary United to the Victimal Priesthood of Her Son

Marie Deluil Martiny presents the Blessed Virgin Mary as the Mother of Christ the Victim Priest and of all who are called to follow Him in the way of His Victimal Priesthood. By her most intimate and perfect participation in the Victimal Priesthood of her Son, the Blessed Virgin Mary is both Coredemptrix and Mediatrix. She is the Virgo sacerdotalis, sacerdotal, not by virtue of sacramental ordination, but by virtue of her unique and entire adhesion to the Sacrifice of Christ. - A Life Offered for Priests

The term that I heard some theologians employing the term of the priesthood of Mary as a ”priesthood by association to the Passion of her Son” on Cross at Calvary. The term has yet to be defined.

Naming Mary's Sacerdotal Role

If the name "priest" cannot be fittingly applied to Mary, by what name should she be called? Mary occupies a unique place between Christ and the rest of the human race. Philosophers tell us whatever is individual is inexpressible. This presents the difficulty of expressing the various functions of Mary in precise words. History records instances when, only after long discussion and multiplied distinctions, was agreement on certain proposed formulas attained. For the uninitiated, the expression "Mother of God" would signify quite normally that Mary gave birth to the Holy Trinity; or the word "co-redemptrix" would indicate that she redeemed us by a title equal to Christ's. It is not surprising then, that a formula sufficiently clear to everyone and capable of stating exactly Mary's role in Christ's priesthood has not been discovered. Since Mary's priestly role cannot be explained either in terms of the priesthood of Christ or in terms of the ordained ministers of the altar, some writers fall back on the "royal priesthood" mentioned by St. Peter. In comparing the disciples of Christ with unbelievers, St. Peter tells them, "You are a chosen race, a royal priesthood, a holy nation, a people of his own."23 This royal priesthood enables all the faithful to offer God "spiritual sacrifice acceptable to God through Jesus Christ."24 This concerns sacrifices in the broad sense of the term, and not with the sacrifice of the body and blood of Jesus Christ. In virtue of their union with Christ in the Mystical Body, all the faithful participate in Him perfections and in Him priesthood. They share in Him priesthood in as much as they offer to God the divine victim, whom the words of consecration uttered by the priest, make present on the altar. They also offer themselves as victims in union with Christ. But they do not participate in it by immolating the divine victim, an act possible only to the ordained minister of Christ at the time of consecration. It is this royal priesthood exercised in the highest degree, these writers conclude, that must be attributed to Mary.

But Laurentin and Neubert ask how anyone can subscribe to such an abas*m*nt of the sacerdotal role of her who was called by God to be Christ's associate in the divine sacrifice. Those writers reduce Mary's role to a common category and forget that she is absolutely unique in her functions. These functions are not identical with those of Christ; they are analogous to them. But neither are they identical with those of other persons. Mary's are of a transcendent order.

What term, then, can express this unique function of Mary? The best and clearest is the phrase derived from the term socia Christi, applied to her by Pseudo-Albert; that is the associate of Christ, our High Priest (socia Christi sacerdotis).

The word socia in itself is not clearer than the word consors, or the word particeps, and could indicate an equality in the priesthood of Christ and of Mary. Some associates are able to occupy the same rank. However, since the time of Pseudo-Albert, the expression socia Christi has a clearly-defined meaning in the history of theology. It designates a secondary action, analogous and united to that of Christ, the action of a "helper like himself." It is equivalent to the phrase employed by Blessed Pius IX, divini sacrificii socia.

The feminine form socia and the allusion to Eve, the first woman and the mother of the living, refer to a priestly role performed by a woman, by a mother. The sacerdotal aspect of Mary's mission consists in being mater et socia Christi sacerdotis. - Priestly Dimension of Mary

Thus it may deduced that there are five manners in which the term priesthood seems to be used within Catholic tradition:

• Aaronic priesthood of the Ancient Israelian tradition.
• Priesthood of Jesus Christ
• Ministerial Priesthood of the Catholic Hierarchy.
• Priesthood of the Faithful
• Priesthood of Mary as yet to be officially defined as socia Christi sacerdotis.