Albert Barnes Commentary Hebrews 7:6

"but he whose genealogy is not counted from them hath taken tithes of Abraham, and hath blessed him that hath the promises." — Hebrews 7:6 (ASV)

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—mh genealogoumenov—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 Hebrews 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 Hebrews 6:12-16.