A royal priesthood
Whether the priesthood of all believers excludes a distinct order of ministers, or is served by one.
1 Peter 2:4-10
4coming to him, a living stone, rejected indeed by men, but chosen by God, precious.
5You also, as living stones, are built up as a spiritual house, to be a holy priesthood, to offer up spiritual sacrifices, acceptable to God through Jesus Christ.
6Because it is contained in Scripture, “Behold, I lay in Zion a chief cornerstone, chosen and precious: He who believes in him will not be disappointed.”
7For you who believe therefore is the honor, but for those who are disobedient, “The stone which the builders rejected has become the chief cornerstone,”
8and, “a stumbling stone and a rock of offense.” For they stumble at the word, being disobedient, to which also they were appointed.
9But you are a chosen race, a royal priesthood, a holy nation, a people for God’s own possession, that you may proclaim the excellence of him who called you out of darkness into his marvelous light.
10In the past, you were not a people, but now are God’s people, who had not obtained mercy, but now have obtained mercy.
What each tradition says
Rabbinic / Midrashic
Later Judaism saw the sanctuary as the manifestation of God's presence among His people. It saw the priest as the vehicle of divine grace and the mediator through whose ministry the sins of the community and of individuals could be atoned for.
Executive Committee of the Editorial Board; Moses Buttenwieser, Priest (The Jewish Encyclopedia, 1901-1906) — In Rabbinical Literature
Patristic
Tertullian holds that the Church’s authority, and the honor sanctified by the joint session of the Order, established the difference between the Order and the laity. Thus where there is no joint session of the ecclesiastical Order, a person offers, baptizes, and is priest alone for himself.
Tertullian, On Exhortation to Chastity (Ante-Nicene Fathers, vol. 4) — Ch. VII
Catholic
Pohle says Catholics also believe in a universal priesthood. That universal priesthood does not exclude a special priesthood; it rather presupposes it. The two are related as general to particular, abstract to concrete, and figurative to real.
Joseph Pohle, The Catholic Encyclopedia, Vol. 12 (1911), article "Priesthood" — "The divine institution of the priesthood" (answer to the Protestant view)
Orthodox
The source says the bishop anoints her head as priests and kings were formerly anointed. This is not because the baptized are ordained priests, but because as Christians they are anointed from Christ the Anointed, a royal priesthood and holy nation, the Church of God.
Anonymous (attributed to Clement of Rome), Constitutions of the Holy Apostles (Ante-Nicene Fathers, vol. 7) — Book III, Sec. II — "On Deacons and Deaconesses, the Rest of the Clergy, and on Baptism"
Reformed / Protestant
Calvin says Moses called your fathers a sacred kingdom because the whole people had royal liberty, and priests were chosen from their body. Now believers are royal priests more excellently, each consecrated in Christ as associates of his kingdom and partakers of his priesthood.
John Calvin, Commentaries on the Catholic Epistles — Commentary on 1 Peter 2:9
Evangelical
Barnes says the priestly business is not entrusted to a particular class known as priests, nor is the name especially given to a particular group. Every Christian is in fact a priest and is engaged in offering an acceptable sacrifice to God.
Albert Barnes, Albert Barnes' Bible Commentary — Comment on 1 Peter 2:5