In the Doctrine and Covenants, section 107 verses 13-17 we read:
13 The second priesthood is called the Priesthood of Aaron, because it
was conferred upon Aaron and his seed, throughout all their
14 Why it is called the lesser priesthood is because it is an
appendage to the greater, or the Melchizedek Priesthood, and has power
in administering outward ordinances.
15 The bishopric is the presidency of this priesthood, and holds the
keys or authority of the same.
16 No man has a legal right to this office, to hold the keys of this
priesthood, except he be a literal descendant of Aaron.
17 But as a high priest of the Melchizedek Priesthood has authority
to officiate in all the lesser offices, he may officiate in the office
of bishop when no literal descendant of Aaron can be found, provided
he is called and set apart and ordained unto this power by the hands
of the Presidency of the Melchizedek Priesthood.
Verses 16 & 17 get to the heart of your question. Only Aaron and his sons have a legal right to the authority. When a descendant of Aaron is not present, then a High Priest in the Melchizedek Priesthood can officiate in the office of bishop, which in LDS theology is the presiding authority in the Aaronic Priesthood.
Today, all worthy young men, starting at age 12 can be ordained to an office in the Aaronic Priesthood. There are a total of four offices: deacon, teacher, priest, and bishop. The worthy 12-years olds are ordained to the office of a deacon. When they turn 14, if they are still worthy, they are ordained to the office of a teacher. At 16 they can be ordained a priest.
Later, (age 18 or so) they can, with approval from their priesthood leaders, be advanced to the Melchizedek Priesthood, and ordained to the office of an elder. This is why LDS missionaries are referred to as "Elder."
The duties of the Aaronic Priesthood are spelled out clearly in the Doctrine and Covenants. We also teach the young men of the church that the purposes of the Aaronic Priesthood are:
- Become converted to the gospel of Jesus Christ and live by its teachings.
- Serve faithfully in priesthood callings and fulfill the responsibilities of priesthood offices.
- Give meaningful service.
- Prepare and live worthily to receive the Melchizedek Priesthood and temple ordinances.
- Prepare to serve an honorable full-time mission.
- Obtain as much education as possible.
- Prepare to become a worthy husband and father.
- Give proper respect to women, girls, and children.
(Taken from Handbook 2: Administering the Church)
As to how long they hold the priesthood, as long as they remain worthy, it is not taken from them. More responsibilities are added as they progress, but "once a deacon, always a deacon."
If you are interested in knowing more about the particular duties of the Aaronic Priesthood in the LDS faith, I would be happy to share them as well. But you probably have to ask that as a separate question.