Thanks for your question.
Genesis chapters 14 gives an account of a battle between the kings of Sodom and Gomorrah, the cities of the plain, and the king of Shinar (Babylon) with three other neighbouring kings. The kings of the area of Sodom were beaten (14:11), and all the inhabitants, their possessions and food were taken. Now this would include Lot, Abraham’s nephew, (14:12), who lived in Sodom, with all his family and possessions.
When Abraham heard of the battle (14:13) he quickly organized an army from his own household, and started off to rescue Lot.
He succeeded in destroying the enemy and returned Lot and all the others to their own places. The king of Sodom was so grateful that he went out to meet Abraham and offered him all the spoil for himself. Abraham refused, saying he wouldn’t even take a shoelace, so that the king of Sodom couldn’t say that he had made Abraham rich. It is on this occasion that Abraham is described as an "Hebrew”. That word had not been used to describe anyone before, but now Abraham and his offspring are identified as Hebrews, meaning "one who crosses over”.
But the interesting part of this story is the introduction of a man, interjected in three verses right in the middle of the narrative, breaking the sequence of thought, seemingly having nothing to do with this story.
This man, Melchizedek, is described as a priest of the Most High God, and King of Salem. He and Abraham have a meal of bread and wine (the first mention of bread and wine in the Bible) together, and as great as Abraham is, Melchizedek blesses Abraham. Abraham gives tithes to Melchizedek. No mention is given as to Melchizedek genealogy or how he came to be a priest, or what happened to him afterwards. This is the first mention of Jerusalem (Salem).
"Then Melchizedek king of Salem brought out bread and wine. He was priest of God Most High, and he blessed Abram, saying, "Blessed be Abram by God Most High, Creator of heaven and earth. And blessed be God Most High, who delivered your enemies into your hand." Then Abram gave him a tenth of everything”. (14:18-20)
Then the curtain is drawn. We hear no more of this man until 1,000 years later. King David in Psalm 110 lifts the curtain again. This psalm is one of the "Messianic psalms”, which prophesy about the coming Messiah. We know from many other passages that it is talking about Jesus Christ.
Psalm 110:4 reads: "The LORD hath sworn and will not repent, Thou art a priest forever after the order of Melchizedek”.
The three verses (Gen. 14:18-20) are picked up in Psalm 110, which in turn is picked up in the New Testament; Matt 22:44; Mark 12:36; Luke 20:42,43; Acts 2:34,35 and particularly in Heb. 1:13.
Jesus Christ is to be a priest after the "order” or "arrangement” of Melchizedek. That is, not of the order or arrangement of the priests of the tribe of Levi under the Law of Moses. Those priests were restricted to being of that tribe, and the High Priest of one family - that of the family of Aaron, (Num. 18:1), and they had to prove their geneology from Aaron. Also under the order of the Levitical priesthood, they performed their services from age 30 to age 50 (Num. 8:25); the High Priest had no age limit, but of course, death would terminate his service anyway.
But it’s not till another 1,000 years had passed, before we get an understanding of what this new order or arrangement of priesthood was all about. Hebrews 7 gives an exposition of this. This chapter presents Jesus Christ as even greater that Melchizedek, who was greater than Abraham.
Firstly the meaning of his name is given - King of Righteousness; then the meaning of the phrase King of Salem - King of Peace. (v. 2).
This is a Bible Principle: That there is no peace until righteousness reigns.
"There is no peace, saith my God, to the wicked” Isa. 57:21
"And the work of righteousness shall be peace; and the effect of righteousness, quietness and assurance forever”. Isa. 32:17
But with Melchisedek’s order no genealogy was established, for there is no mention of his parents, though of course he would have had some. So, therefore, his "order” includes King-priests "out of every nation, kindred, tongue and people” (Rev. 5:9,10). The believers of Peter’s day are called "a royal priesthood” (1 Peter 2:9).
"without father, without mother, without descent (genealogy R.V)” (v. 3).
"having neither beginning of days nor end of life” (v. 3)
Concerning Melchizedek, no mention is made of his death in the Genesis record, so he stands in the Word of God as an eternal priest. Of course he literally died, as all men do, but what the Scriptures did not say about this man speaks loud and clear.
"made like unto the Son of God” (v. 3) By this is meant that the record that Scripture has of this man’s life, brief though it may be, contains all that is necessary to set Melchizedek out as a type of the priesthood of Messiah. (Mostly by what it does not say). The rest of Heb. 7 continues the exposition of the change of priesthood to Jesus Christ.
v. 4-10 Inherent greatness of Melchizedek above other priests.
v. 11-19 Levitical priesthood superseded
v. 20-22 Christ appointed priest by oath
v. 23-25 His unchangeable priesthood
v. 26-28 Christ’s fitness as High Priest
I hope you have found this helpful.