The Melchizedec Priesthood

In the Restored Church of  Jesus Christ of Latter-day Saints authority is held by the Melchizedec Priesthood, the highest and holiest priesthood, and is after the order of the Son of God.  All other authorities are appendages to this priesthood.  It is evident that the prophets of old held the priesthood,  but not the Aaronic Priesthood as this was for making sacrifices and held by the descendants of Aaron.  Joseph Smith said: "All the prophets held the Melchizedec Priesthood and were ordained by God himself," meaning Jehovah.  These persons so honoured held their authority by special dispensation, as the other priesthood order was the Levitical, as mentioned above.

When Christ came, he being a high priest forever after the order of Melchizedec, (Heb.2:17-18) the order held by the prophets was once again spread forth among the people:
16 Ye have not chosen me, but I have ^achosen you, and ^bordainedyou, that ye should go and bring forth ^cfruit, and that your fruit should remain: that whatsoever ye shall ask of the Father in my^dname, he may give it you.  John 15:16
5 Ye also, as ^alively stones, are built up a ^bspiritual ^chouse, an holy priesthood, to offer up ^dspiritual ^esacrifices, acceptable to God by Jesus Christ.
9 But ye are a ^achosen generation, a ^broyal ^cpriesthood, an ^dholy^enation, a ^fpeculiar people; that ye should shew forth the praises of him who hath called you out of ^gdarkness into his marvellous^hlight:   1 Peter 2:5,9

So the Melchizedec Priesthood was restored, under apostolic direction a kingdom of priests being again found on the earth.  However, after the apostles had all died there was no one left holding the keys to authorise a person to be ordained to any priestly office, and in this manner the Lord took the priesthood  from the Earth. ( See Rev.12)

he Melchizedek Priesthood was first made known to Adam, and the patriarchs and prophets in every dispensation had this authority (D&C 84:6–17; Teachings of the Prophet Joseph Smith, pp. 180–81). When the children of Israel failed to live up to the privileges and covenants of the Melchizedek Priesthood, the Lord took away the higher law and gave them a lesser priesthood and a lesser law. These were called the Aaronic Priesthood and the law of Moses. The Aaronic Priesthood is not a different priesthood; rather, it is the lesser portion of the priesthood, dealing with the introductory ordinances and the preparatory commandments (D&C 84:18–28). When Jesus came, he restored the Melchizedek Priesthood to the Jews and began to build up the Church among them. However, it was lost again by apostasy, and was taken from the earth.

The Melchizedek Priesthood was restored to the earth in these last days by the ministry of Peter, James, and John, who literally came to Joseph Smith and Oliver Cowdery in the spring of 1829 and conferred this power and authority upon them (D&C 27:12–13). Later, Moses, Elias, and Elijah gave them further keys by which these brethren could use the Melchizedek Priesthood in additional ways (D&C 110). The president of The Church of Jesus Christ of Latter-day Saints is the president of the high or Melchizedek Priesthood, and by virtue of this position, he holds all the keys that pertain to the kingdom of God on the earth. This office or calling is held by only one man at a time, and he is the only person on the earth at that time in whom all the powers and keys of the Melchizedek Priesthood are functional.