What exactly is a priest?
Let's look at the definition of priest, according to the dictionary (American Heritage by Houghton Mifflin, 2000):
priest [priːst] feminine, priestess
In Biblical Judaism, priests came from the tribe of Levi. They were the ones who served at the temple. The High Priest was the head honcho, the one who went in to the Holy of Holies once a year to atone for his own sins and the sins of Israel. He was the mediator between man and God. The Hebrew word is Kohen. Today, the Jewish surnames such as Cohen, Cohn, Levine, Levy... they all reflect back to the priestly tribe of Levi.
The first mention of a priest in scripture, however, was Melchizedek, king of Salem, in Genesis 14:18, which is actually a preincarnate appearance of the Messiah. The next mention of a priest is in Genesis 41:45, where Joseph marries the daughter of a priest of On in Egypt. There are numerous other references to these Egyptian priests. The next mention comes when Moses is taking a wife in Midian - her father Jethro was a priest. After the Levitical priesthood was established in the Moses Covenant, the scriptures make hundreds and hundreds of references to the Levitical priests, and to pagan priests as well (priests of Ba'al, for example).
After the temple was destroyed in 70 AD, the Judaic priesthood was abolished along with the temple sacrifices. Since then, the Jewish faith has had rabbis and sages, but no priests. The early believers in Yeshua did not have priests, either. (Paganism, however, continued to have priests and priestesses at their temples that were scattered throughout the Roman Empire and all over the world.)
As that early body of believers grew, the congregations began to appoint leaders to their fellowships. Several words are used for this leadership, and they are synonymous and used interchangeably. The word bishop comes from the Greek word episkopouv, meaning overseer. The Greek word presbureros means elder. The word for pastor - poiemen - means shepherd. Look at Acts 20:28, in which Paul was addressing the elders (presbureros) of Ephesus: Therefore take heed to yourselves and to all the flock, among which the Holy Spirit has made you overseers (episkopos), to shepherd (poimaino) the church of God which He purchased with His own blood. This is not a hierarchy of church leadership (ie: priest, bishop, archbishop, cardinal, pope). This is a group of godly leaders - protecting, guiding, and discipling their local fellowship, with Yeshua as the head.
The entire book of Hebrews gives a wonderful description of the priesthood, and explains how Yeshua has now fulfilled this office for us. The curtain was torn, and we now have direct access to Almighty God through Him, our mediator and High Priest.
There are three more references to the priesthood in scripture after the book of Hebrews; two in 1 Peter 2, and one in Revelation 1:6. These do not refer to church leadership, but to US! Listen to this:
1 Peter 2:9 But you are a chosen generation, a royal priesthood, a holy nation, His own special people, that you may proclaim the praises of Him who called you out of darkness into His marvelous light.
ALL of us as believers are to be priests... that is, mediators between God and a lost and dying world! Living stones, according to verse 5, being built up into a spiritual household!
So why do some churches still distinguish between clergy and laity, the clergy being the priests? In the 4th century, the Roman Emperor Constantine had a vision which led to him declaring Christianity as the legal, state religion of the Roman Empire. All the pagan temples that were scattered about became "Christian," as did the priests that served there. Thus, there was a "resurrecting" of the priesthood for church leadership. Much more could be said on that subject, and in fact, my posts under the label seven churches have a lot of information regarding church history if you wish to read further.