Hebrews 5:8-10 (ESV)
Although he was a son, he learned obedience through what he suffered.  And being made perfect, he became the source of eternal salvation to all who obey him,  being designated by God a high priest after the order of Melchizedek..
The priesthood that God established for the people of Israel was a picture of the work that Jesus would do in offering himself to God on our behalf. Hebrews 10:1 calls the priesthood of Aaron, a shadow of the reality that is in Christ. Jesus was always the real priest.
Nevertheless, the requirement for being a priest was that one be from the tribe of Levi. Jesus was from the tribe of Judah. Some could argue that Jesus’ priesthood is not legitimate since He is from the wrong tribe. The answer to such a complaint is that Jesus is a priest in a priesthood that predates Aaron by a long shot. He is a priest in the order of Melchizedek.
We read about Melchizedek in Genesis and the Messiah is prophesied to be a priest in the order of Melchizedek in Psalm 110:4. The author’s point is that the priesthood of Jesus is not based on Aaron. It is based on something greater than Aaron.
How is Jesus like Melchizedek? He is greater than Abraham. The priesthood of Jesus cannot be denied on the basis of tribal disqualification. Jesus is greater then such a requirement.
How does all this mysterious Melchizedek stuff apply to us? Our Saviour predates everything. He is not the One who fits in with the Law. The Law is that which matches up with Him. He is more than just the fulfilment of Old Testament prophecy. He is the One who already existed when the prophecies were written. He is the reason the prophecies came into being in the first place. Melchizedek, like everything else in the Old Testament, paints a picture of Christ. Christ is greater. And it is Christ you have if you are trusting Him. Who could ask for anything more?