Albert Barnes on Hebrews 7:6

But he whose descent is not counted from them. Melchizedek. The word descent is, in the margin, pedigree. The meaning is, that he was not in the same genealogy--μηγενεαλογουμενος--he was not of the order of Levitical priests. That Melchizedek is meant there can be no doubt; at the same time, also, the thought is presented with prominence, on which Paul so much insists, that he was of a different order from the Levitical priesthood.

And blessed him. Blessed him as a priest of God; blessed him in such a manner as to imply acknowledged superiority. See Heb 7:1.

That had the promises. The promise that he should have a numerous posterity; that in him all the nations of the earth should be blessed. See Heb 6:12-16.

(1) "descent" "pedigree" (b) "tithes" Gen 14:20 (c) "the promises" Rom 9:4