Again, the author goes back to the idea of Jesus as a Jewish high priest; specifically, "a priest forever according to the order of Melchizedek."
In case you're not up on your obscure biblical figures, that's a reference to a guy who appears briefly in Genesis to bless Abraham. He's also mentioned in one line of the Psalms. Seriously, blink and you'll miss the dude.
Why is this Melchizedek so important?
Well, the author explains that Melchizedek was a guy whose name means "king of righteousness" and "king of peace." He was never born and he never died (weird), but on the plus side, that means he got to stay a priest forever.
In Leviticus, God appointed descendants of Jacob's son Levi as the new priests of his tabernacle—and the people were supposed to tithe to them (i.e., pay them ten percent of their earnings).
In Genesis, Melchizedek blessed Abraham and Abraham gave him ten percent of his money.
Jewish law came down while the Levite priests were in power. But that law didn't do anything to stop sin, did it? In fact, says Hebrews, Psalm 110 points out that there would be another high priest coming. One who was just like Melchizedek. One who wasn't part of the Levi's tribe at all. One who would fix everything.
Sound like anyone you might know?
Now, Jesus wasn't a Levite. He was an ancestor of Jacob's son Judah (just like King David). No one from Judah's tribe has ever served as high priest, but the author knows that the high priest that God appoints is going to be just like Melchizedek. He's not gonna come by bloodlines, but through the power of God himself.
See, says the author, Jewish law was imperfect. But Jesus is perfect, and he's come to bring the world a new and better covenant with God.
He won't be like the old priests either who were always dying off and changing. Jesus gets to hold onto his position forever, which means anyone who comes to him can be saved at any time.
Jesus also never sinned, so he doesn't have to offer sacrifices for his own bad deeds like human high priests. He doesn't have to keep burning rams and goats at the altar for everyone else because he made himself into a sacrifice by dying on the cross. His work here is done.