The Melchizedek Priesthood, also known as the "higher" priesthood, is named after the ancient patriarch and high priest Melchizedek, to whom Abraham paid tithes. Doctrine and Covenants 107:3-4 explains why the priesthood is called after him:
Before his [Melchizedek's] day it was called the Holy Priesthood, after the Order of the Son of God. But out of respect or reverence to the name of the Supreme Being, to avoid the too frequent repetition of his name, they, the church, in ancient days, called that priesthood after Melchizedek, or the Melchizedek Priesthood.
The Melchizedek Priesthood was given to Adam and has since been held by every patriarch and prophet authorized by God. This priesthood was taken from Israel as a whole during the time of Moses. Instead, Israel was given the lesser portion known as the Aaronic Priesthood. Christ restored the Melchizedek Priesthood to His apostles. After the death of the apostles, the priesthood was lost during what is known as the Great Apostasy. In 1829, Christ's apostles then restored the Melchizedek priesthood to Joseph Smith and Oliver Cowdery.
Offices and Functions
Just as in the Aaronic Priesthood, a man retains all the keys and authority of his previous ordinations. For example, if no Aaronic Priesthood members are present, the Melchizedek Priesthood can bless and pass the sacrament.
Elder as a Title
The title "Elder" can be used for any holder of the Melchizedek Priesthood. As a title it is typically used to refer to male Mormon missionaries (e.g. Elder Smith or Elder Jones) or to General Authorities (such as Elder Russell M. Nelson or Elder Merrill J. Bateman).
Elder as an Office
Men are ordained as elders beginning at either age 18 or 19. The duties of an elder are to baptize, confirm members of the Church and give the gift of the Holy Ghost, administer the sacrament, give blessings of comfort and healing, and generally watch over the Church.
Members of the Quorums of the Seventy are called as General Authorities under the direction of the First Presidency and the Quorum of the Twelve Apostles. The Seventy direct missionary work and other Church activities, build up the Church in all the nations the Church is in, and preach the Gospel. Members of the Seventy may also be called upon to speak in General Conference. At present, there are five quorums, each with no more than seventy members. Members of the Seventy may be honorably released and given emeritus status.
High Priests have the authority to officiate in the Church. Men are ordained as High Priests to serve in a number of callings: members of the bishopric, members of the stake presidency, high councilors, mission presidents, stake patriarchs, etc. High Priests have their own quorums, separate from the Elder's quorum.
Patriarchs are called on stake levels and are ordained by General Authorities or stake presidents given that authority by the Quorum of the Twelve Apostles. A patriarch's duty is to give worthy Church members special blessings known as patriarchal blessings. Such blessings are the Lord's personal words to the recipient and may give the person a better understanding of their callings in life. The office of patriarchs is held for life, though if the patriarch is no longer able to function in his duties, an additional stake patriarch may be called. The term "patriarch" is also applied to the father of a family.
Meaning "one sent forth," apostles are special witnesses of Jesus Christ and are called for life. The men ordained as apostles are members of either the Quorum of the Twelve Apostles or the First Presidency. The apostles travel throughout the world building up and regulating the Church. Each apostle is given all the keys of the kingdom, but only the senior apostle--the President of the Church--is authorized to use all the keys. The other apostles act under the president's direction.
President of the Church
At any one time, only one man can hold the office of President of the Church and exercise the keys and authority associated with that office. The President (also referred to as "the Prophet") is the senior apostle (in terms of years as an apostle, not age. See Choosing a Prophet to learn about the Order of Succession) and is also the president of the Melchizedek Priesthood. He is sustained as the prophet, seer, and revelator. As such he is entitled to receive revelation for not only the Church, but the entire world. A man will remain President of the Church the remainder of his life, though in cases of poor health the counselors in the First Presidency may take over some of the functions of the office as the President authorizes.
Authority of the Melchizedek Priesthood
The Melchizedek Priesthood "holds the right of presidency, and has power and authority over all the offices in the church in all ages of the world, to administer in spiritual things," and holds "keys of all the spiritual blessings of the church" (D&C 107:8, 18).