Psalm 110’s allusion to Melchizedek is a reference back to a short story from Abraham’s time. After war broke out across the land (seemingly between humans and the giant descendants of the Nephilim) and Abraham’s nephew Lot was kidnapped, Abraham headed out with his servants to rescue him. He entered into the war and won, which is where Melchizedek comes into the story.
And Melchizedek king of Salem brought out bread and wine. (He was priest of God Most High.) And he blessed him and said,
“Blessed be Abram by God Most High,
Possessor of heaven and earth;
and blessed be God Most High,
who has delivered your enemies into your hand!” (Ge 14:18-20)
There’s a lot we could say about all of this, but for now our focus is on the fact that Melchizedek was the last king/priest we have in the Old Testament. Adam seemed to first serve in a role like this, but Melchizedek is the last glimpse we get of such a figure until we get to Jesus.
And that is the point of Psalm 110: Jesus will be a king/priest forever after the order of Melchizedek—the most recent king/priest. Jesus will take on the role forever and restore the role that God always intended to exist. Jesus has become King of the world because,
God has highly exalted him and bestowed on him the name that is above every name, so that at the name of Jesus every knee should bow, in heaven and on earth and under the earth, and every tongue confess that Jesus Christ is Lord, to the glory of God the Father. (Philip 2:9-11)
And as for the priest side of things in Jesus’ identity? Well, Psalm 110 was not the only passage after Genesis to reference Melchizedek. Hebrews, likewise makes comment and connects Jesus straight to Melchizedek and the priesthood.
We have this as a sure and steadfast anchor of the soul, a hope that enters into the inner place behind the curtain, where Jesus has gone as a forerunner on our behalf, having become a high priest forever after the order of Melchizedek. (He 6:19-20)
Melchizedek became an important character in the Jewish literature that was written closer to Jesus’ time. He became a Messiah-esque figure that the Jews began to ponder about, wondering who he was going to be based on the prophetic Psalm about him. Much of their thinking becomes very flawed, but Hebrews shows us the true meaning that Melchizedek served in pointing to Jesus; for Jesus is the new priest for all eternity. Jesus picks up the role that Melchizedek had last and no one will ever take it from Him.