Melchizidek appears in the Book of Genesis, arriving mid-story, without prior introduction, hosts a parley between Abraham and the King of Sodom, provides bread and wine- and gives Abraham his blessing. He is described as King of Salem (meaning Peace) and priest of the Most High God (the first person in the Torah to be given the title "cohen") and everybody seems to defer to him.
He has haunted the Judaeo-Christian imagination ever since. Tradition says he never died. He has been identified as the Messiah, The Holy Spirit, an Avatar of Jesus, Jesus himself....
The essential thing about him is that he stands outside any sacerdotal lineage or initiatory tradition. He is inherently priestly. To be a "priest in the order of Melchizidek" is to owe your priesthood to no-one but God.
I've filched this picture from Wikipedia. It's by the Netherlandish painter Dieric Bouts and depicts the meeting of Abraham and Melchizidek. Abraham ( a sprightly 71 at the time of the encounter) is the guy in armour.