The author of Hebrews talked about Melchizedek and Levi to describe Jesus’ work as a priest.
God established the Jewish priestly system in the Law of Moses. Men from the tribe of Levi served as priests. Men from the family line of Aaron served as high priest. When a high priest died, another man from Aaron’s family line took his place.
Yet verse 4 of Psalm 110 announced that the Messiah would be a priest for ever just like Melchizedek. Melchizedek wasn’t part of the Jewish priestly system. He wasn’t from Levi’s tribe. Melchizedek’s family line isn’t recorded in the Old Testament. For the author of Hebrews, this meant that Melchizedek had no beginning and no end. In this way it’s understood that his service as a priest lasts for ever.
Jesus wasn’t from the tribe of Levi either. As God’s Son, Jesus has no beginning and no end. His service as a priest also lasts for ever. Jesus didn’t become a priest because the Jewish priestly system allowed him to. He became a priest because God promised he would be. He became a priest because he has powerful eternal life that death can’t destroy.
In this way he completes the Jewish priestly system. No one else needs to serve as a priest between God and people. No more sin offerings need to be made for people to be forgiven. Jesus’ sacrifice saves people from the power of sin for ever.