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A priest in the LDS Church is a priesthood office to which young men are ordained at the age of 16, or adults who are still progressing to higher priesthood offices. Most men in the church will be ordained priests after ecclesiastical interviews with their leaders and through demonstrating faithful living. From D&C 20:
60 Every elder, priest, teacher, or deacon is to be ordained according to the gifts and callings of God unto him; and he is to be ordained by the power of the Holy Ghost, which is in the one who ordains him.
A priest has responsibilities in the church to administer the sacrament, see that meetings are held regularly and in accordance with the Spirit, and to help take care of physical needs of the congregation:
46 The priestβs duty is to preach, teach, expound, exhort, and baptize, and administer the sacrament,
47 And visit the house of each member, and exhort them to pray vocally and in secret and attend to all family duties.
48 And he may also ordain other priests, teachers, and deacons.
49 And he is to take the lead of meetings when there is no elder present;
50 But when there is an elder present, he is only to preach, teach, expound, exhort, and baptize,
51 And visit the house of each member, exhorting them to pray vocally and in secret and attend to all family duties.
52 In all these duties the priest is to assist the elder if occasion requires.
Oh, and priests are also typically educated in seminary, in areas where it is available. In most parts of the world, members who are high school age go to seminary class before or during school. Its a four year program of scripture and doctrine study. These classes are also for the young women.
Unlike other Christian denominations, a priest does not necessarily give a sermon on Sundays. Speakers are chosen by the local priesthood leaders and can be nearly anyone in the ward (local congregation).
I'm mobile now but if required I can cite some scriptures.
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In short, it's related to the fact that The Church of Jesus Christ of Latter-day Saints has a lay ministry.
With only very few exceptions, all leaders and teachers in the LDS Church serve in a part-time capacity.
Most members have an official role or "calling," but regardless of their specific current calling, eligible members are ordained to various offices in the priesthood.
For example, there is usually only one Elders Quorum President at a time in a ward (congregation), but almost all 18+ men will receive the Priesthood and be ordained an Elder.
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An extra answer to Matt's answer to provide extra details.
Priest is a priesthood office to which typically faithful young men are ordained when they turn 16 and male adult new converts are ordained shortly after their baptism and confirmation. Ordination to the next highest priesthood office, elder, typically occurs as or shortly after faithful young men enter adulthood or, for faithful male adult converts, a year or so after their baptism and confirmation.
All LDS congregations I have been a member of or have heard of have more elders than priests, usually far more. This is because the number of active priests in a congregation is usually limited to those young men who are 16 or 17 years old and adult male members who have been members for less than a year, while the number of elders is usually most of the adult men. (As an FYI there is another, higher priesthood office for adult men, "high priest", but there are usually more elders than high priests in a congregation.) Thus, I am curious where you got your statement that "usually almost half of the congregation is priests."
The seminary Matt references in his answer is not a prerequisite to become a priest (or any other priesthood office), and in fact since the seminary course typically runs from ages 14-18 it is most common for young men to be ordained priests when they turn 16 and have not yet completed the full seminary course.
Priests are able to baptize but not confirm, can ordain deacons, teachers (priesthood offices lower than priest), and priests, can bless the sacrament in addition to passing the sacrament (which deacons and teachers can also do) and preparing the sacrament (which teachers can also do).
For more details on what the duty is for priests in the LDS church, please reference Doctrine & Covenants 20:46-60, noting that the duties of the teachers and deacons are included in the duties of priests. Since the scripture passage is fairly long and is only ancillary to the OP's question, I do not quote it here.