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Disclaimer: My knowledge of "priesthood of all believers" is limited, what I write always reflects my understanding of the general doctrine and may differ between denominations.
Priesthood of all believers (as far as I understand):
All believers have the priesthood, meaning they can, for example baptise. No ordination required, the authority comes from the Holy Ghost.
LDS:
Priesthood is conferred by ordination of someone holding authority to do so (another priesthood holder). All worthy men may receive the priesthood, and are encouraged to do so. Priesthood also entails a convenant with God. Many priesthood holders have a list that traces back who ordained whom and can trace back their priesthood to Joseph Smith (and then, through Joseph Smith, to Peter, James and John and ultimately Jesus Christ himself).
Priesthood of all believers
While all lay people are priests, most congregations have professional ministers. Other functions are filled by the members of the congregation. Degree of central organization differs. Lutherans are very organized, Evangelicals supposedly less so. Since everyone is a priest, it is unclear if the priesthood entails any duties.
LDS
The priesthood entails well-defined duties (notably the ones in D&C 20) which express themselves in the congregational life, for example that every priesthood holder (except deacons) gets the responsibility to regularly visit and care for a number of member families (home teaching). The priesthood is also the reason why a mission is expected of a young adult man, but not of women (which can still go but don't share the same duty).
Everyone is serving in the church one way or another, the specific calling is given by the leadership (the bishop in case of a ward calling), and understood to be inspired. With the exception of Apostles, all callings end after a few years, at which point the person is assigned a different calling. Everyone is lay, including bishops and all higher positions. Anecdotally, my different bishops' professions over the years have included everything from bus driver to university professor.
So, the similarity is certainly that almost all men encountered in the church are priesthood holders. The difference is where the authority to act in God's name (the priesthood) is coming from. Churches which believe in "priesthood of all believers" derive their priesthood from the Holy Ghost. LDS believe that this is not enough, it must be transferred from someone that already has the authority to act in God's name.
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This address given by Elder Dallin H. Oaks of the Quorum of the Twelve Apostles of the LDS Church addresses just this. I would encourage you to read it. I will summarize it here.
Priesthood ordination is given only to male members of the Church. Of these, only a subset are called as leaders that possess priesthood keys, which keys are what God has given to govern and direct the use of priesthood on the earth. Ordinances such as baptism, confirmation, administration of the sacrament, etc. are performed by those who have had the priesthood conferred on them under the direction of those with priesthood keys, with the exception of a certain ordinance that occurs in the temple that is performed by women for women without those women performing the ordinance being ordained to the priesthood.
What I just described is different from "priesthood of all believers" as I understand the concept. What I am about to describe, again from the address from Elder Oaks, I believe is quite similar to the "priesthood of all believers."
The authority of the priesthood belongs to all those who are members of the church --- men, women, and children --- within the scope in which they have been given responsibility. Thus, the Relief Society president (the Relief Society is the LDS Church's organization for women) holds priesthood authority with respect to the women in her congregation, teachers hold priesthood authority to teach the word of God in their respective classes, etc. I personally would extrapolate this to say that all members of the LDS Church hold priesthood authority since all are called by God to share the gospel with those whom they come in contact, and in that sense there is a type of "priesthood of all believers" in the LDS Church, though I don't think it exactly matches with how the term is normally used.