Melchizedek was a great Old Testament high priest, prophet, and leader who lived after the flood and during the time of Abraham. He was called the king of Salem (Jerusalem), king of peace, king of righteousness (which is the Hebrew meaning of Melchizedek), and priest of the most high God.
Abraham paid tithes to Melchizedek (Genesis 14: 18-20). Melchizedek’s people wrought righteousness, and obtained heaven (Joseph Smith Translation, Genesis 14: 25-40). Christ was a high priest after the order of Melchizedek (Hebrews 5: 6). Melchizedek was King of Salem, priest of the most high God (Hebrews 7: 1-3). None were greater than Melchizedek (Alma 13: 14-19). Abraham received the priesthood from Melchizedek (Doctrine and Covenants 84: 14). In respect for the Lord’s name, the ancient Church called the higher priesthood the Melchizedek Priesthood (Doctrine and Covenants 107: 1-4). 
- Melchizedek was a notable prophet and leader who lived about 2000 B.C. He is called the king of Salem (Jerusalem), king of peace, and “priest of the most High God.” Unfortunately, information concerning him in the Bible is relatively scarce, being limited to Genesis 14: 18-20; Hebrews 4: 6; and Hebrews 7: 1-3. Mention of the priesthood of Melchizedek is given in several other instances, primarily in Psalms and in Hebrews. However, latter-day revelation gives us much more about him and his priesthood (see Joseph Smith Translation, Genesis 14: 17-40; Joseph Smith Translation, Hebrews 7: 1-3; Alma 13: 14-19; Doctrine and Covenants 84: 14; Doctrine and Covenants 107: 1-4). From these sources we realize something of the greatness of this prophet and the grandeur of his ministry. 
Melchizedek and Shem
Shem lived five hundred years after fathering Arkpasad, and then died at the age of six hundred (Genesis 11:10, 11). Therefore, his death took place thirteen years after the death of Sarah (1881 B.C.) and ten years after Rebecca and Isaac married (1878 B.C.) In that light, it has been opined that it is possible that Shem might have been Melchizedek (which does translate to King of Righteousness), the priest-king to whom Abraham paid tithes (Genesis 14: 18-20). This interpretation was supported by Jewish midrashim (ancient stories and commentaries). 
- So righteous and faithful was Melchizedek in the execution of his high priestly duties that he became a prototype of Jesus Christ (Hebrews 7:15). The Book of Mormon prophet Alma said of him, "Now, there were many [high priests] before him, and also there were many afterwards, but none were greater" (Alma 13:19). The Doctrine and Covenants states that Melchizedek was "such a great high priest" that the higher priesthood was called after his name. "Before his 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 the ancient days, called that priesthood after Melchizedek, or the Melchizedek Priesthood" (Doctrine and Covenants 107:2-4; italics in original). 
Worthy male members of The Church of Jesus Christ of Latter-day Saints hold the priesthood of Melchizedek, the authority to act in the name of God and the authority to seal ordinances for this life and the next.
From Joseph Smith's Commentary on the Bible, compiled and edited by Kent P. Jackson, we learn, under the notation on Gen. 14:18:
"It is understood by many...that Melchizedek was king of some country or nation on the earth. But it was not so. In the original it reads "king of Shalom," which signifies king of peace or righteousness, and not of any country or nation. (Discourse of 27 August 1843, recorded by James Burgess; WJS, 246.)
"The word "Salem" is a wrong translation. It should be "Shalom," signifying peace. (Discourse of 27 August 1843, recorded by William Clayton, WJS, 247.)
"Salem" is designed for a Hebrew term. It should be "Shalom," which signifies righteousness and peace. As it is, it is nothing--neither Hebrew, Greek, Latin, French, or any other." (Discourse of 27 August 1843, recorded by Willard Richards; WJS, 244.)
In Lectures on Faith, Joseph states clearly and unequivocally that Shem and Melchizedek are indeed one and the same person.
Additional note: The name Shem is where we get the word Semite or Semitic. It comes from the more proper Shemite or Shemitic.