What authority did the Priests of the Old Covenant have, and how was it perfected in the New Covenant? In other words, how does the priesthood of the Old Covenant and the New Covenant compare, especially regarding authority in teaching faith and morals?
In the Old Testament we see a three-fold priesthood. There is the common or universal priesthood of all Israelites at the bottom (Ex. 19:6), a ministerial priesthood above them (Ex. 19:22, 24, Lev. 1:5), and a high priest at the top (Num. 35: 25). We find a similar three-fold priesthood under the New Covenant as well. There is the common or universal priesthood of all Christians (1 Peter 2:5,9), a ministerial priesthood above them (Rom. 15:16), and a high priest at the top (Heb. 3:1).
The New Testament verses cited for the priesthood of all believers are based on a passage from the Old Testament (Ex. 19:6). If you look that up, you will find that it refers to an Old Testament priesthood; not to the ministerial priesthood that served at the Tabernacle or Temple but to a priesthood of all Israelites. This priesthood is clearly not the same as the ministerial one because in the same chapter God tells the Israelites that they are all priests. He also speaks of “the priests who come near to the Lord” (Ex. 19:22) and warns that “the priest and the people” must not be allowed to go up on the sacred mountain that Moses does (Ex. 19:24). This speaks of the priests and the people as two different groups, showing that while all Israelites are priests in one sense, only some are priests in another sense – a point that is forcefully demonstrated in the rebellion of Korah (Num. 16), where God makes the point in an unmistakable way.
So, each type of priesthood had authority only to the extent that was appropriate to its level. In New Testament terms, this would mean that the ordinary Christian (part of the universal priesthood) cannot be a priest in the strict sense, for he can offer only the figurative sacrifice of prayer. Only the next level, the ministrial priesthood, may offer the sacrifice of the Mass for the people. The high priest is Christ. According to the Catholic Encyclopedia, "The typical character of the high-priest is explained by St. Paul (Heb., ix), where the Apostle shows that while the high-priest entered the “Holy of Holies” once a year with the blood of victims, Christ, the great high-priest, offered up His own blood and entered into Heaven itself, where He “also maketh intercession for us” (Rom., viii, 34; . . .).