What is the Melchizedek priesthood?

In biblical Christianity, the Melchizedek priesthood is an office that applies only to Christ. Melchizedek is introduced in Genesis 14:18 and is described as the king of Salem and “priest of God.” Abram (later Abraham) offers Melchizedek a tithe and is blessed. The name Melchizedek is the combination of the Hebrew words for “king” and “righteous,” making Melchizedek a righteous, kingly priest.
In Matthew 22 Jesus debates the Pharisees. In verse 44 Jesus cites Psalm 110:1, stating that the Messiah is David’s “lord” in that verse. Melchizedek is mentioned in the same psalm: “The Lord has sworn and will not change his mind: ‘You are a priest forever, in the order of Melchizedek’” (Psalm 110:4). So the Messiah, David’s lord, was given Melchizedek’s priesthood. Melchizedek is therefore a type of Christ—some commentators say Melchizedek’s appearance to Abraham was actually a Christophany, an early revelation of Jesus Christ.
Much of chapters 6 and 7 of the book of Hebrews is given to explaining why Jesus’ Melchizedek priesthood is superior to that of Aaron. Hebrews 7:23–24 implies that Jesus holds His priestly office eternally, using a Greek word that suggests something that cannot be passed down or changed. Hebrews 7:26 calls this priest “exalted” and sinless.
For all of these reasons, Christ alone can fulfill the office of the Melchizedek priesthood (Hebrews 6:20), at least according to the Bible.
Mormonism holds a different, unbiblical view of the priesthood of Melchizedek. According to Mormon doctrine, the office the Melchizedek priesthood stopped until it was re-instituted through the ministry of Joseph Smith (Doctrine and Covenants 107:1–5). Mormonism teaches that men may be ordained into this priesthood, through offices such as Apostle, Patriarch, or Elder. The invoking of Melchizedek and, to a lesser extent, Aaron, is used by Mormonism to arrogate priestly authority for their offices.