Albert Barnes on Hebrews 7:16

Who is made. That is, the other priest is made--to wit, the Messiah. He was made a priest by a peculiar law.

Not after the law of a carnal commandment. Not according to the law of a commandment pertaining to the flesh. The word carnal means fleshly; and the idea is, that the law under which the priests of the old dispensation were made was external, rather than spiritual; it related more to outward observances than to the keeping of the heart. That this was the nature of the Mosaic ritual in the main, it was impossible to doubt, and the apostle proceeds to argue from this undeniable truth.

But after the power of an endless life. By an authority of endless duration, That is, it was not concerned mainly with outward observances, and did not pass over from one to another by death, but was unchanging in its character, and spiritual in its nature. It was enduring and perpetual as a priesthood, and was thus far exalted above the service performed by the priests under the former dispensation.