Overview of the Old Testament Priesthood

The following quotes are taken from the Catholic Encyclopedia entry on "Priesthood," which I think are worth sharing. I will quote the article freely, with slight edits for readability and keeping things brief.

The word priest is derived from the Greek presbyteros (the elder). By the term is meant a (male) person called to the immediate service of the Deity and authorized to hold public worship, especially to offer sacrifice.

In the age of the Patriarchs the offering of sacrifices was the function of the father or head of the family (cf. Genesis 8:20; 12:7, etc.; Job 1:5). Hummelauer's hypothesis that this pre-Mosaic priesthood was abolished in punishment of the worship of the golden calf (Exodus 32) can hardly be scientifically established.

In the Mosaic priesthood we must distinguish: priests, Levites, and high-priest.

Priests. It was only after the Sinaitical legislation that the Israelitic priesthood became a special class in the community. From the tribe of Levi, Jahweh chose the house of Aaron to discharge permanently and exclusively all the religious functions; Aaron himself and later the first-born of his family was to stand at the head of this priesthood as high-priest, while the other Levites were to act, not as priests, but as assistants and servants. The solemn consecration of the Aaronites to the priesthood took place at the same time as the anointing of Aaron as high-priest (Exodus 29:1-37; 40:12 sqq.; Leviticus 8:1-36). 

The official duties of the priests related partly to their main occupations, and partly to subsidiary services. To the former category belonged all functions connected with the public worship, e.g. the offering of incense twice daily (Exodus 30:7), the weekly renewal of the loaves of proposition on the golden table (Leviticus 24:9), the daily offering of the morning and evening sacrifices, especially of the lambs (Exodus 29:38 sqq.). As subsidiary services the priests had to sound the trumpets announcing the holy-days (Numbers 10:1 sqq.), declare the lepers clean or unclean (Leviticus 13-14; Deuteronomy 24:8; cf. Matthew 8:4).

The priests furthermore were teachers and judges; not only were they to explain the law to the people (Leviticus 10:11; Deuteronomy 33:10) without remuneration (Micah 3:11), but they had also to settle difficult lawsuits among the people (Deuteronomy 17:8; 19:17; 21:5). 

 
Levites in the narrow sense. As simple servants of the priests, the Levites might not enter the sanctuary, nor perform the real sacrificial act, especially the sprinkling of the blood. This was the privilege of the priests (Numbers 18:3, 19 sqq.; 18:6). The Levites had however to assist the latter during the sacred services, prepare the different oblations and keep the sacred vessels in proper condition. In common with the priests, the Levites were also bound to instruct the people in the Law (2 Chronicles 17:8; Nehemiah 8:7).

The high priest. At Jahweh's command Moses consecrated his brother Aaron first high-priest. The consecration of Aaron consisted in washings, investment with costly vestments, anointing with holy oil, and the offerings of various sacrifices (Exodus 29). The high-priest was for the Jews the highest embodiment of theocracy, the monarch of the whole priesthood, the special mediator between God and the People of the Covenant, and the spiritual head of the synagogue. He presided at the Sanhedrin. He alone could perform the liturgy on the Feast of Expiation.

With the destruction of Jerusalem by Titus in A.D. 70 the entire sacrificial service and with it the Jewish priesthood ceased. The later rabbis never represent themselves as priests, but merely as teachers of the law.

The Christian priesthood. When the Prophet Isaias foresees the entrance of pagans [Gentiles] into the Messianic Kingdom, he makes the calling of priests from the heathen (i.e. the non-Jews) a special characteristic of the new Church (Isaiah 66:21): "And I will take of them to be priests and Levites, saith the Lord". Now this non-Jewish (Christian) priesthood in the future Messianic Church presupposes a permanent sacrifice, namely that "clean oblation", which from the rising of the sun even to the going down is to be offered to the Lord of hosts among the Gentiles (Malachi 1:11). The sacrifice of bread and wine offered by Melchisedech (cf. Genesis 14:18 sqq.), the prototype of Christ (cf. Psalm 109:4; Hebrews 5:5 sq.; 7:1 sqq.), also refers prophetically, not only to the Last Supper, but also to its everlasting repetition in commemoration of the Sacrifice of the Cross (see MASS). Rightly, therefore, does the Council of Trent emphasize the intimate connection between the Sacrifice of the Mass and the priesthood: "Sacrifice and priesthood are by Divine ordinance so inseparable that they are found together under all laws. Since therefore in the New Testament the Catholic Church has received from the Lord's institution the holy visible sacrifice of the Eucharist it must also be admitted that in the Church there is a new, visible and external priesthood into which the older priesthood has been changed." Surely this logic admits of no reply.

Some things I'd like to highlight about the above: 

• The term 'priest' comes from the Greek word for "Elder" (Presbyter), which is the same as saying Firstborn, since the first born child is the oldest. I would argue that the "Elders" of the New Testament have some claim to the Priesthood (in the original Genesis sense), and thus the NT presupposes that the "Apostles and Elders" (parallels to "chief priests and elders" throughout the Gospels) were priests, along with their teaching and pastoral duties.
  
• The Patriarch (head of the family), was the Father, and throughout Genesis we see these Fathers offering Sacrifices (e.g. Noah, Gen 8:20; Abraham, Gen 12:7; Isaac, Gen 26:25; Jacob, Gen 33:20; Job 1:5). It is unclear what happened to these priests/sacrifices, since the OT shifts to the Levitical Priesthood. The early chapters of Leviticus say that a layman could bring an offering and slaughter it before giving it to the priests, so maybe this could be the link between the two priesthoods. This article denies the theory that the Patriarchal priesthood was lost due to the sin of the Golden Calf.
  
• Among the Tribe of Levi, there were three orders of clergy: High Priest, Priest, and Levite. Not all Levites were priests, only the family line of Aaron were priests. The rest of the Levites acted basically like Deacons.
  
• Despite the fact the Levitical Priesthood had restricted the priesthood to the biological ancestors of Aaron, the Prophet Isaiah foresees in the future New Covenant that God will call some of the Gentiles to be His priests and Levites (deacons). This is quite astonishing, for the last thing the Israelites imagined would be Gentile Priests and Gentile Deacons. It can only mean that God wasn't planning on the Aaronic Priesthood to last forever. This beautiful prophecy seems to only be fulfilled in the Catholic Church (and Eastern Orthodoxy), not Protestantism.
  
• The article notes that after the destruction of the Temple, the Jewish priesthood ceased. From then on, their leaders only referred to themselves as Rabbis, that is teachers. Sounds similar to Protestants, how they only refer to their leaders as Pastor, Doctor, Preacher, etc, and avoid terms like Father, Priest, etc. 

Questions to ponder; answers to seek out:

• It seems that those not of Aaron's house could offer sacrifices or perform priestly duties. For example, Moses sprinkles blood to ratify the Covenant (Ex 24:4-8). The Jewish Encyclopedia gives a list of men who sacrificed without being of the Tribe of Levi, so there must have been two priesthoods existing alongside one another:
  

  In ancient Israel one was not required to be specially consecrated in order to perform the sacrificial functions; any one might approach the altar and offer sacrifices. Thus Gideon, of the tribe of Manasseh (Judges 6:26 et seq.), and the Danite Manoah (Judges 13:16,19) sacrificed in person at the express command of God and the angel of God respectively; similarly, David sacrificed on the altar he had built at God's command (2 Sam 24:25); and Solomon, before the ark in Jerusalem (1 Kings 3:15). David, on the occasion of the transference of the Ark to Zion, and Solomon, at the dedication of the Temple of Jerusalem, ministered as priests (2 Sam. 6:14,17,18; 1 Kings 8:22,54 el seq.); the latter continued to personally offer sacrifices on the altar of Yhwh at regular intervals (1 Kings 9:25). Similar instances, in later times, are presented by Elijah, sacrificing on Mount Carmel (1 Kings 18:32 et seq.), and by Ahaz, in the Temple at Jerusalem (2 Kings 16:12 et seq.). Even where there was a sanctuary with a priesthood, as at Shiloh, any layman might slaughter and offer his sacrifices without priestly aid (comp. 1 Sam. 2:13-16). The Ephraimite Samuel became priest of the sanctuary at Shiloh, wearing the priestly linen coat and the pallium (1 Sam. 2:18 et seq., 1 Sam 3:1). The kings of Israel ordained as priest whomever they chose (1 Kings 41:31); David, too, invested his own sons, as well as the Jairite Ira, of the tribe of Manasseh, with the priestly office (2 Sam. 8:18; 20:26). After the capture of its ark by the Philistines the sanctuary of Shiloh disappeared from history; its priesthood, however, appeared in the following period at the sanctuary of Nob, which also had an ephod (1 Sam. 14:3; 21:1,10; 22:9,11).

• It is not clear why Aaron was chosen, though he was firstborn in his family, elder brother of Moses. 
• It is not clear what the Levites attained in their ordination of Exodus 32, unless it was to signify that all the Tribes originally could serve as Deacons, alongside Aaron's sons.