Hebrews 7:1-3 meaning

There is not much written about Melchizedek in the Bible besides here, Psalm 110 and in Genesis 14:18-20. There are several things mentioned about Melchizedek: he was the king of Salem, priest of the Most High God, who met Abraham as he was returning from the slaughter of the kings, he blessed Abraham (v 1), and he received a tithe from Abraham (to whom also Abraham apportioned a tenth part of all the spoils) (v 2). Abraham had mustered over three hundred of his servants to rescue his nephew, Lot, from captivity. After successfully freeing Lot and regaining stolen goods, Abraham apportioned a tenth part of all the spoils to the king of Salem: Melchizedek. 

In this passage, the Pauline Author will draw parallels between Melchizedek and Christ showing that Melchizedek is a greater priest than the Levites, and so Christ is greater as well. Melchizedek was first of all, by the translation of his name, king of righteousness, and then also king of Salem, which is king of peace (v 2). Melchizedek's name means king and righteousness, Salem means peace, so his name means king of righteousness and king of peace—representative of Christ.

The Pauline Author writes that, in the biblical account, Melchizedek was without father, without mother, without genealogy, having neither beginning of days nor end of life (v 3). No genealogy was recorded in the Bible for Melchizedek because he was not a priest on the basis of his lineage as the Levites were, just as Christ was named a priest after the order of Melchizedek, apart from the lineage of the Levites (Psalm 110:4). 

The Pauline Author points out that Melchizedek's death was also not recorded in the Bible which is representative of the true eternal nature of Christ as our high priest.

Melchizedek's priesthood is greater than the Levites. This is apparent first from the fact that Abraham gave him a tithe, demonstrating that Melchizedek is a greater authority than Abraham. This would be significant to the Jewish people since Abraham was one of the great fathers of their faith. 

It demonstrates there is an order of priesthood that is higher in authority than those appointed from Abraham's decedents. Furthermore—as the Pauline Author will explain in the next verses—the Levites were born out of Abraham's seed. In a way, the lesser priests, the Levites, paid a tithe to Melchizedek.

The point is that Melchizedek is a representation of our high priest, Christ: a king-priest of righteousness and of peace. Made like the Son of God, He remains a priest perpetually (v 3). And His authority is above the authority of any other priest.