The Lord makes this promise on oath and will not revoke it: “You are an eternal priest after the pattern of Melchizedek.”
Who is Melchizedek?
We only read of Melchizedek once in the Bible:
Then Melchizedek king of Salem brought out bread and wine. He was priest of God Most High, and he blessed Abram, saying, “Blessed be Abram by God Most High, Creator of heaven and earth. And praise be to God Most High, who delivered your enemies into your hand.” Then Abram gave him a tenth of everything.
After this we read of him again in Psalm 110 and Hebrews 7. Both of these following passages don't tell us more about Melchizedek, but rather tells us something about Jesus. There is therefore very little known about Melchizedek, other than that he was a priest before the time of Aaron, and that he was the king of Salem (early Jerusalem).
What does it mean to say someone is a priest from the order of Melchizedek?
Jesus is considered a priest in the order of Melchizedek because, like Melchizedek, Jesus was not a descendant of Aaron, and thus would not qualify for the Jewish priesthood under the Law of Moses.
Pulpit commentary notes - David, admitted into the councils of the Most High, has been made aware that the Messiah is, by God's decree, to be both King and Priest. God has "sworn" this, and will certainly not draw back from his oath. Thou art a priest forever after the order of Melchizedek. Not, like ordinary priests, a priest for a few years, or for a lifetime, but a priest forever and ever (לעולם) - seeing "he ever liveth to make intercession for us" (Hebrews 7:25). And a priest "after the order of Melchizedek." Not, that is, after the order of Aaron, who was a priest and nothing more, but after that of Melchizedek, the elder priesthood, which combined the offices of priest and king (see Hebrews 5:6-10; Hebrews 7:1-10, 20-28).
What does it mean to say that the Messiah will be a priest?
GotQuestions writes - Like the Levitical priests, Jesus offered a sacrifice to satisfy the Law of God when He offered Himself for our sins (Hebrews 7:26-27). Unlike the Levitical priests, who had to continually offer sacrifices, Jesus only had to offer His sacrifice once, gaining eternal redemption for all who come to God through Him (Hebrews 9:12).