Objection 1. It would seem unfitting that Christ should be a priest. For a priest is less than an angel; whence it is written (Zechariah 3:1): "The Lord showed me the high-priest standing before the angel of the Lord." But Christ is greater than the angels, according to Hebrews 1:4: "Being made so much better than the angels, as He hath inherited a more excellent name than they." Therefore it is unfitting that Christ should be a priest.
Objection 2. Further, things which were in the Old Testament were figures of Christ, according to Colossians 2:17: "Which are a shadow of things to come, but the body is Christ's." But Christ was not descended from the priests of the Old Law, for the Apostle says (Hebrews 7:14): "It is evident that our Lord sprang out of Judah, in which tribe Moses spoke nothing concerning priests." Therefore it is not fitting that Christ should be a priest.
Objection 3. Further, in the Old Law, which is a figure of Christ, the lawgivers and the priests were distinct: wherefore the Lord said to Moses the lawgiver (Exodus 28:1): "Take unto thee Aaron, thy brother... that he [Vulgate: 'they'] may minister to Me in the priest's office." But Christ is the giver of the New Law, according to Jeremiah 31:33: "I will give My law in their bowels." Therefore it is unfitting that Christ should be a priest.
On the contrary, It is written (Hebrews 4:14): "We have [Vulgate: 'Having'] therefore a great high-priest that hath passed into the heavens, Jesus, the Son of God."
I answer that, The office proper to a priest is to be a mediator between God and the people: to wit, inasmuch as He bestows Divine things on the people, wherefore "sacerdos" [priest] means a giver of sacred things [sacra dans, according to Malachi 2:7: "They shall seek the law at his," i.e. the priest's, "mouth"; and again, forasmuch as he offers up the people's prayers to God, and, in a manner, makes satisfaction to God for their sins; wherefore the Apostle says (Hebrews 5:1): "Every high-priest taken from among men is ordained for men in the things that appertain to God, that he may offer up gifts and sacrifices for sins." Now this is most befitting to Christ. For through Him are gifts bestowed on men, according to 2 Peter 1:4: "By Whom" (i.e. Christ) "He hath given us most great and precious promises, that by these you may be made partakers of the Divine Nature." Moreover, He reconciled the human race to God, according to Colossians 1:19-20: "In Him" (i.e. Christ) "it hath well pleased (the Father) that all fulness should dwell, and through Him to reconcile all things unto Himself." Therefore it is most fitting that Christ should be a priest.
Reply to Objection 1. Hierarchical power appertains to the angels, inasmuch as they also are between God and man, as Dionysius explains (Coel. Hier. ix), so that the priest himself, as being between God and man, is called an angel, according to Malachi 2:7: "He is the angel of the Lord of hosts." Now Christ was greater than the angels, not only in His Godhead, but also in His humanity, as having the fulness of grace and glory. Wherefore also He had the hierarchical or priestly power in a higher degree than the angels, so that even the angels were ministers of His priesthood, according to Matthew 4:11: "Angels came and ministered unto Him." But, in regard to His passibility, He "was made a little lower than the angels," as the Apostle says (Hebrews 2:9): and thus He was conformed to those wayfarers who are ordained to the priesthood.
Reply to Objection 2. As Damascene says (De Fide Orth. iii, 26): "What is like in every particular must be, of course, identical, and not a copy." Since, therefore, the priesthood of the Old Law was a figure of the priesthood of Christ, He did not wish to be born of the stock of the figurative priests, that it might be made clear that His priesthood is not quite the same as theirs, but differs therefrom as truth from figure.
Reply to Objection 3. As stated above (7, 7, ad 1), other men have certain graces distributed among them: but Christ, as being the Head of all, has the perfection of all graces. Wherefore, as to others, one is a lawgiver, another is a priest, another is a king; but all these concur in Christ, as the fount of all grace. Hence it is written (Isaiah 33:22): "The Lord is our Judge, the Lord is our law-giver, the Lord is our King: He will" come and "save us."
Source: Summa Theologica (New Advent)