What does Hebrews 7:3 mean?
The writer of Hebrews is using the figure of Melchizedek to explain how Jesus Christ offers us a superior covenant with God. As part of that, he seeks to show that Melchizedek is greater than Abraham: Abraham was blessed by Melchizedek and offered Melchizedek tithes. The reference to the priest figure in this verse is this same Melchizedek who the Old Testament describes as king of Salem—meaning "peace"—and as a priest of God Most High (Genesis 14:18).The Greek phrase used here is a figure of speech and not meant to be taken absolutely literally. By saying that Melchizedek is "without father or mother or genealogy," the writer simply means that his lineage is unknown: he is agenealogētos. This turn of phrase meant a person of unknown or obscure birth, possibly even of low birth. This point is used symbolically by the author of Hebrews, as a parallel to Jesus Christ and His eternality.
As it applies to this section, this relates to the nature of the priesthood. Human priests come and go—they are born, they age, and they die. Their priesthood cannot continue forever. Melchizedek, then, as someone with no recorded beginning or end, serves as a metaphor for the priesthood which God promised: one without end, in a priesthood which lasts forever.