There is no Christian priest (hiereus) in the New Testament. Saint Paul pays tribute to more than a dozen Christian ministries, but none of them is the priesthood. He never calls himself or his assistants priests, and never offers sacrifice (the priestly act). Jesus was a layman, not a priest. He did not even belong to the priestly line of Levi. But he went to the Temple where priests offered sacrifice -- and so did his early followers. James the brother of Jesus kept the first Christians of Jerusalem observant of Temple worship. He directs Paul to get himself ritually purified from travel in the Temple, and Paul does so (Acts 2:17-24). So, until the destruction of the Temple in 70 C.E., Jerusalem Christians had the same (and only) priests as other Jews.
Then where did a separate Christian priesthood come from? At the end of the first century C.E., a group (probably in Rome) of Christians missed the comfort of Jewish worship some of them had experienced before the year 70. The unidentifiable author of the New Testament's "Epistle to Hebrews" assured them, in elegant Greek, that the old Jewish rites were useless anyway, since Jesus was a better priest with a better sacrifice. The Jews had just offered animal sacrifice. Jesus offered human sacrifice (a savage act in most cultures), since he was both priest and victim. Was -- not is -- since, according to Hebrews 10:11-15, he offered himself only once, in an unrepeatable way, while Jewish priests repeated their ineffectual offerings.
Though Jesus could not be a Jewish priest, since he was not a Levite, this author said he was a priest like the mythical Melchizedek, a priest of Canaanite gods to whom Abraham paid a tithe. Later, Catholic priests would claim descent from Melchizedek (though he had no descendents) and claim to repeat the sacrifice of Jesus (though "Hebrews" says it was "once for all").
Many religions have sacrificial rites, and priests to offer them. Some early Christians obviously felt the jibes of their contemporaries that they had no sacrificial buildings and no sacrificing priests. So Christians acquired both. They had to make the body and blood of their communal meal become a real body for a real sacrifice, though this meant that the physical body of Jesus was in many places at once, hiding under the substance-less "accidents" of bread and wine. This made Jesus relive (or re-die) on altars his agony on the cross.
A long line of intellectual Christians, typified by Saint Augustine, denied that sacrifice and consumption of the body of Jesus was an original part of the religion. A typical passage in Augustine is from his Sermon 227:
What you see [bread and wine] passes away, but what is invisibly symbolized does not pass away. It perdures. The visible is received, eaten, and digested. But can the body of Christ be digested? Can the church of Christ be digested? Can Christ's limbs be digested? Of course not.
The claim that the body of Christ was being sacrificed on an altar is not in the earliest liturgies of the Christian meal, which were "thanksgiving" meetings (eucharist is, etymologically, "giving thanks").
Even while the author of "Hebrews" was dismissing the Jewish priesthood, some people kept yearning back to it, and adopted features of it. The Christian priesthood became, like the Jewish priesthood, all and only male, and male without blemish. Thomas Aquinas said that the Christian priesthood had adopted the purity rules for the Jewish priesthood (Leviticus 21:26-24), but added abstention from sex as an even higher holiness code (Summa Theologiae 3a.36 a 3, 39a6). Thus, the priests who were absent from early Christianity became the monopolizers of "true" Christianity in Roman and Eastern rites.
Some Christians, like the Anglicans, have and honor their own priests; but popes have told them these are not real priests, since they do not descend from the mythical Roman bishopric of Peter. In dismissing other people who do things in the name of Jesus, the Vatican resembles the Apostle John, returning with the disciples Jesus had sent out on their first mission:
"Master," said John, "we saw a man driving out devils in your name, but as he is not one of us we tried to stop him." Jesus said to him, "Do not stop him, for he who is not against you is on your side." (Luke 9:49-50).
We live in a time when Catholic priests are an aging and shrinking group, damaged in morale and reputation, overstretched in their monopolization of all sacramental services. Already, lay deacons and catechizers and readers, instructors for baptism and marriage, are filling in for the diminished priestly ranks in Catholic parishes. Some think the clerical shortage will be solved by recruiting new people for the priesthood -- married priests, women priests, gay priests. When we run out of everyone else, will we start ordaining child priests? Anything to keep the sacrificing priesthood?
What we really need are no priests. We should remember what Jesus told the disciples in Matthew 23:8-11:
You must not be called "rabbi"; for you have one Rabbi, and you are all brothers. Do not call any man on earth "father"; for you have one Father, and he is in heaven. Nor must you be called "teacher"; you have one Teacher, the Messiah.
When Jesus told his disciples not to call themselves rabbis or fathers or teachers, he did not add that they should not call themselves priests. No one had yet imagined that there might be Christian priests.