The Melchizedek priesthood is a combination of the kingship and priesthood. Both Josephus and Jasher teach the first priest/King was Adam and the government was passed down to Noah. Sometime after the Flood Shem settled and built (or rebuilt if it existed before the flood) the city of Salem (later to be renamed Jerusalem) on the mountains of Moriah. (Morah means teacher in Hebrew) Shem started the School of the Lord at Salem. All who wanted to learn the ways of the Lord were welcome to come to the city and the school. The patriarchs Abraham, Isaac, and Jacob all attended there. (compare this to the school of the prophets ran by the prophet Elijah)

During this time the sons of Canaan, son of Ham, settled into the land of Canaan. Shem tried to change the hearts of the sons of Canaan toward the Lord but most rejected the Lord and turned to magic and false gods. Shem ruled over 14 nations in relative peace. The title of the king priest of Salem was Melchizedek meaning king of righteousness, and also known by the title of Adonizedek meaning Lord or righteousness.

Shem was the Melchizedekian priest that blessed Abraham after he had returned from slaying Chedorlaomer and rescuing Lot. Shem passed the Melchizedekian priesthood on to Abraham, then Issac, then Jacob. Jacob divided the Melchizedekian priesthood into 2 parts; giving the Kingship to Judah and the priesthood to Levi. At this point the Melchizedekian priesthood was no more and the Levitical priesthood began.

We can See By Joshua 10 that after the death of Shem the school was corrupted by the Canaanites to the point that the Amorite King took the title of Adonizedek and reigned from Jerusalem, replacing the teaching and worship or the one true God for Baal and Astoreth. Joshua destroyed the Amorite King and his army but the Jebusites retained control of Jerusalem until the time of David.

This split of king and priest continued until the time when Jesus (of the Kingly line of Judah) was baptized into the priesthood by John the Baptist; thereby reestablishing the Melchizedekian priesthood. See Hebrews 7 for further details.

Hebrews 7
Paul here is comparing the Melchizedekian priesthood with the Levitical priesthood. To be a Levitical priest one had to be of the genealogy of Aaron through Levi. one had to start training at age 20, but could not be a full priest till the age of 30 and there was a mandatory retirement at the age of 50. Jesus was ordained in the order of Melchizedek which has no genealogy or end of life. therefore Jesus will remain our priest forever. Paul is showing the priesthood of Jesus is superior to the priesthood of Levi.

Related Legends
One Rabbinical legend says Shem inherited the coats of skins that Adam, showing he was the rightful ruler; this however contradicts the teaching in Jasher that they were stolen by Nimrod.

Jasher seems to indicate that Noah was there with Shem, at Salem, till his death.

Information taken from church father Ephrem the Syrian, Book of Jasher, Josephus, Targum Onqelos, Targum Pseudo-Jonathan and other ancient Jewish works