JOEL - I have most often heard it refered to as the "Priesthood of all believers".
Many of the Protestant religions believe that through baptism and a simple belief in Christ, all Christians are accepted into the priesthood of Christ. One of the main scriptures they get this idea from is in 1 Peter 2:1-10:

1. "WHEREFORE laying aside all malice, and all guile, and hypocrisies, and envies, and all evil speakings,
2. As newborn babes, desire the sincere milk of the word, that ye may grow thereby:
3. If so be ye have tasted that the Lord [is] gracious.
4. To whom coming, [as unto] a living stone, disallowed indeed of men, but chosen of God, [and] precious,
5. Ye also, as lively stones, are built up a spiritual house, an holy priesthood, to offer up spiritual sacrifices, acceptable to God by Jesus Christ.
6. Wherefore also it is contained in the scripture, Behold, I lay in Sion a chief corner stone, elect, precious: and he that believeth on him shall not be confounded.
7. Unto you therefore which believe [he is] precious: but unto them which be disobedient, the stone which the builders disallowed, the same is made the head of the corner,
8. And a stone of stumbling, and a rock of offence, [even to them] which stumble at the word, being disobedient: whereunto also they were appointed.
9. But ye [are] a chosen generation, a royal priesthood, an holy nation, a peculiar people; that ye should shew forth the praises of him who hath called you out of darkness into his marvellous light:
10. Which in time past [were] not a people, but [are] now the people of God: which had not obtained mercy, but now have obtained mercy." (1 Pet 2:1-10)

and:

5. And from Jesus Christ, [who is] the faithful witness, [and] the first begotten of the dead, and the prince of the kings of the earth. Unto him that loved us, and washed us from our sins in his own blood,
6. And hath made us kings and priests unto God and his Father; to him [be] glory and dominion for ever and ever. Amen. (Rev 1:5-6)

Even though these scriptures don't say exactly how one obtains the priesthood, they are nevertheless interpreted as meaning that all you have to do is believe in Christ and through His atonement all are holders of the "royal priesthood".
Martin Luther insisted that everyone who trusts in Jesus Christ is a priest. The concept that all who believe in Christ are priests occurred to him after he became convinced that Scripture was the only authority for a Christian. As he studied the Bible he decided that in and through Jesus Christ a believer possessed the righteousness of God, and therefore, immediate access to God without the mediation of an arrogant priesthood. (Luther's Primary Works , ed. Henry Wace and C. A. Bucheim (London: Hodder and Stoughton, 1896), 399.)

John Calvin wrote, "Christ . . . once for all offered a sacrifice of eternal expiation and reconciliation; now, having also entered the sanctuary of heaven, he intercedes for us. In him we are all priests (Rev. 1:6; cf. 1 Peter 2:9) (Calvin: Institutes of the Christian Religion, Vol. 2, ed. John T. McNeill, trans. and index. Ford Lewis Battles (Philadelphia: The Westminster Press, 1960, 1476)).

Many Protestants believe that in likening the whole body of believers to the priesthood of ancient Israel, it removes the possibility of a spiritual aristocracy or hierarchy within Christianity. It makes all equal in the sight of God so no individual is set above others. According to them we do not need human priests to represent us before God.
It is true that all are equal in the sight of God, Him being no respector of person, but when Jesus set up His church:

"He gave some, apostles; and some, prophets; and some, evangelists; and some, pastors and teachers;
For the perfecting of the saints, for the work of the ministry, for the edifying of the body of Christ:" (Eph 4:11-12)

So apparently some sort of priesthood leadership organization is necessary for Christ's true church.
The scriptures say that Jesus, "ordained twelve, that they should be with him, and that he might send them forth to preach, (Mark 3:14)
Christ also told his apostles:

"Ye have not chosen me, but I have chosen you, and ordained you, that ye should go and bring forth fruit, and that your fruit should remain: that whatsoever ye shall ask of the Father in my name, he may give it you.

And He specifically told Peter:

"And I will give unto thee the keys of the kingdom of heaven: and whatsoever thou shalt bind on earth shall be bound in heaven: and whatsoever thou shalt loose on earth shall be loosed in heaven." (Matthew 16:19)

When the apostles of Jesus, ... "had ordained them elders in every church, and had prayed with fasting, they commended them to the Lord, on whom they believed." (Acts 14:23)

These acts of ordinations and bestowal of keys in Christ's ancient church were certainly different from what the the common believer in Christ would have received at that time. If we were to have Christ's church today it is logical that we would still need the same priesthood authority, ordinations, and church offices.
The difference comes in the traditional Christian's interpretation of the scriptures, trying to figure out how one obtains God's authority. They really have to stretch the interpretation of a few verses to make them mean what they want.
Besides the few scriptures I quoted above it isn't easy to defend against it because there isn't much more in the Bible that talks about it. Most of what we know comes through latter-day revelation, which specifies how priesthood is bestowed on man and how it operates.(D&C 13, 20, 84)