I recently encountered a thread on a Catholic forum wherein some were taking umbrage with the Latter-day Saint claim that faithful male members hold the Melchizedek Priesthood. While there are important differences, it should be noted that, in Catholic theology, ordained sacerdotal priests participate in the Melchizedek Priesthood that Jesus holds. In the 1994 Catechism of the Catholic Church, we read the following:
. . . The presentation of the offerings at the altar takes up the gesture of Melchizedek and commits the Creator's gifts into the hands of Christ who, in his sacrifice, brings to perfection all human attempts to offer sacrifices. (¶ 1350)
Everything that the priesthood of the Old Covenant prefigured finds its fulfillment in Christ Jesus, the "one mediator between God and men" The Christian tradition considers Melchizedek, "priest of God Most High," as a prefiguration of the priesthood of Christ, the unique "high priest after the order of Melchizedek" . . . (¶ 1544)
Commenting on the sacrament of Holy Orders in Catholic theology, Jesuit priest and scholar F.M. De Zulueta wrote the following:
The possession of true Orders, on the other hand, is quite a different matter. Once these are received, the powers attached to them can never be taken away. Just as the true priest shares in Christ’s Priesthood, so does he share in its perpetuity: ‘Thou art a priest for ever according to the order of Melchisedec.’ The word—spoken of Christ—has its application in a subordinate sense to His ministering priests. (F.M. De Zulueta, Letters on Christian Doctrine (Second Series), Part II, Volume III [5th ed.; London: Burns Oates and Washbourne Ltd, 1921], 60-61)
For more on the priesthood, be sure to check out my book:
