JOEL - Actually there really isn't anything specifically called the "Mormon Levitical Priesthood", although the "Aaronic Priesthood" is sometimes referred to by this term.(Heb. 7:11; D&C 107:1, 6, 10).
Our Doctrine and Covenants states that "there are, in the church, two priesthoods, namely, the Melchizedek and Aaronic, including the Levitical Priesthood" (D&C 107:1). The Aaronic Priesthood, was instituted under the Law of Moses at the time when Israel rejected the responsibilities of the higher or Melchizedek Priesthood. God gave them a "lesser priesthood" comprising specific areas of authority dealing with sacrifices and temporal concerns of salvation (Ex. 20:19; JST). This authority was granted as a right to Aaron and his lineal descendants forever. Levitical Priesthood refers to certain duties within the Aaronic Priesthood that were delegated to worthy male members of the tribe of Levi. Therefore, in the strict sense the Levitical Priesthood is a lesser part of the Aaronic Priesthood, held among those who were Levites, but not of the family of Aaron. It is anticipated that in the restoration of all things, the sons of Levi will once again function in the Levitical Priesthood on the earth (Mal. 3:2-3).
Only direct descendants of Aaron could be ordained priests. The firstborn among the sons of Aaron would preside over the other priests. This type of priesthood organization and service continued in Israel until the coming of Jesus Christ.
The Aaronic priesthood is the authority by which John the Baptist (a descendant of Aaron) prepared the way for Jesus Christ, teaching faith, repentance, and baptism for the remission of sins (Matt. 3:1-17; Mark 1:1-11; Luke 1:5-80; John 1:15-34; Acts 8:14-17; D&C 84:25-28).
After Jesus was baptized by John He confered upon His apostles the higher or Melchezidek priesthood(John 15:16, Mark 3:14). Following his death, resurrection, and ascension, Christ continued to direct his Church by giving commandments to the apostles through the power of the Holy Ghost (Acts 1:2) and through the authority of the higher Melchizedek Priesthood that he had conferred upon them. After the death of the apostles there followed a general apostasy, during which many gospel principles were lost and all the powers of the priesthood were withdrawn from the earth (2 Thes. 2:1-4; 2 Tim. 3:1-5).
On May 15, 1829, John the Baptist appeared to Joseph Smith and Oliver Cowdery as a resurrected messenger from God and conferred the ancient "Priesthood of Aaron" upon them (D&C 13). Later, the Apostles Peter, James and John appeared to them and confered the higher or Melchezidek priesthood.
In these latter-days, the Aaronic Priesthood has not been restricted to those who are literal descendants of Aaron or of Levi, since those lineages are not at present identified. Also that priesthood authority that implemented the ordinances of the Law of Moses has been replaced by the higher priesthood(Melchezidek) and laws and ordinances of the gospel of Jesus Christ. Today the Aaronic priesthood is often referred to as the "preparitory priesthood" and serves a slightly different function than it did in ancient times. It is confered upon young men between the ages of 12 and 18 and gives them experiences in serving God and the Church by performing certain ordinances such as baptism and administration of the sacrament. This experience prepares them to receive the higher priesthood(Melchezidek) at age 19 where additional authority, responsibilities and opportunities for service are placed upon them.
(Much of this information was taken from The Encyclopedia of Mormonism)