Levitical priesthood vs Catholic priesthood
By Luke Lancaster
Within the Old Covenant of Judaism, there were twelve tribes of Israelites. One of those tribes was chosen by God to serve Him as priests, known as the Levitical priesthood. There was a three-fold hierarchy to them: The high priest, the priests, and the Levites. Besides this, all twelve tribes were considered to be a "kingdom of priests" (Exodus 19:6). This three-fold hierarchy was continued in the New Covenant, with Christ as the high priest, priests, and deacons. The rest of the Church was considered a "kingdom of priests" as well (1 Peter 2:9).
The first high priest of the Old Covenant was Aaron, Moses's brother (Exodus 28:1). Every year, the high priest would enter into the God's presence and offer a sacrifice. Within Judaism, God was worshipped in the Temple. The Temple had a holy place, and a holy of holies. God's presence was within the holy of holies, and the high priest alone could enter into that sacred place. Jesus functioned as the last high priest, and He entered God's presence to offer the sacrifice of Himself (Hebrews 2:17).
Old Covenant Lower Priests
The first set of priests that served under the high priest was Aaron's sons (Ex. 28:1). They were chosen and called out from the people of Israel to enter the holy place and offer sacrifices to God. The Levites would hear the sins of the people, and God would work through the priests (Leviticus 5:4-6). They were "consecrated" (Ex. 30:30), but first they were washed with water (Lev. 8:6).
New Covenant Lower Priests
The apostles were priests. They were chosen and called out from the disciples of Jesus (Luke 6:13). The apostles were commissioned to offer the sacrifice of bread and wine to God (Lk. 22:19-20). They were to hear the sins of the people and to either offer God's forgiveness or deny God's forgiveness to them (John 20:21-23). The apostles were "consecrated" (John 17:17-19), but first they were washed with water (Jn. 13:5).
Old Covenant and New Covenant Levites
The Levites who were not descended from Aaron were charged with assisting the priests in the outer courtyard of the Temple. Similarly, the apostles chose seven deacons to assist them (Acts 6:1-6). Soon, the deacons began assisting at the sacrifice of the Mass, so that both the Old and New Covenant Levites assisted in sacrifice.
This is a very generalized way of looking at the connection to how God set up holy orders in the Old and New Covenants. God was served through high priests, priests, and Levites. The rest of the people of God were considered a kingdom of priests dedicated to God and offering their lives to Him. If interested in analyzing each particular point, see the other articles on this website.