Several people have recently asked me about the mysterious figure of Melchizedek and why the book of Hebrews makes such a big deal out of him, even though he’s only mentioned in 4 verses in the whole Old Testament! So for anyone else looking to get some insight into the significance of this enigmatic ancient priest-king, here are my notes on him:
First appears in Genesis 14:17-20, where he is described as king of Salem and priest of “God Most High” (El Elyon).
Salem is to be identified with Jerusalem, which was occupied by the Jebusites before David took it over (see Josh 15:63; 2 Sam 5:6-10).  Some scholars understand Melchizedek’s dual office of priest and king of Jerusalem as something that was passed on in the city’s Jebusite culture. We later see both David and Solomon perform priestly roles even though they were not Levites (2 Sam 6:14-19; 1 Kgs 8), likely because as kings of Jerusalem they were continuing this tradition of the king sharing a priestly role, as well. 
Melchizedek blesses Abram and receives a tribute from him (a tithe, or tenth). They also share a meal of bread and wine together. These actions indicate the making of a covenant or peace treaty between Abram (who had just won a substantial military victory against invading kings) and Melchizedek, who was now perhaps the most influential king in the region after the conclusion of that conflict. 
The next reference to Melchizedek is in Psalm 110:1-4, which is a messianic psalm (meaning it focuses on the Davidic kingly line). It describes God issuing a promise to David’s heir that he will rule until God subdues all his enemies, and that he is a “priest forever according to the order of Melchizedek.” This is probably another reference to that dual-office of priest-king typified in Melchizedek, now affirmed as part of God’s covenant with David’s line.
The fact that Davidic kings are called “priests according to the order of Melchizedek” in Psalm 110 became an important point for early Christians, who saw Christ as their High Priest but had to reckon with the fact that he wasn’t from the tribe of Levi. Psalm 110 is one of the most frequently-quoted psalms in the New Testament. This theme of the Melchizedekian priesthood is taken up in Hebrews 7 to explain how Christ’s priesthood is superior to the Levitical priesthood.
Hebrews 7:1-22 (NRSV):
This “King Melchizedek of Salem, priest of the Most High God, met Abraham as he was returning from defeating the kings and blessed him”; 2 and to him Abraham apportioned “one-tenth of everything.” His name, in the first place, means “king of righteousness”; next he is also king of Salem, that is, “king of peace.” 3 Without father, without mother, without genealogy, having neither beginning of days nor end of life, but resembling the Son of God, he remains a priest forever.
The author of Hebrews makes a big deal out of the lack of genealogy for Melchizedek in Genesis to show that one could be a priest appointed by God even without ancestral qualifications. This is exactly the point he’s going to build on with regard to Christ’s priesthood.
In contrast to all the other main characters in Genesis having genealogies and notes about their parentage and deaths, Melchizedek has neither but still receives blessing from the patriarch Abraham himself. The author of Hebrews sees Melchizedek’s uniqueness as a foreshadowing of Christ’s. He’s essentially viewing Melchizedek’s significance retroactively through the lens of Christ’s priesthood.  Such typological interpretation might seem fanciful to us, but first-century Jews didn’t have a problem interpreting with such methods. This argument would have impressed them.
It’s important to note that the author of Hebrews says Melchizedek resembles the Son of God. There have been a number of early Church Fathers and other interpreters who thought Melchizedek was an appearance of the pre-incarnate Christ, but Hebrews seems to make a strong distinction between the two men. Melchizedek is simply like Christ by virtue of his unique priestly office and in the air of “timelessness” about him that our author sees in the text of Genesis.
4 See how great he is! Even Abraham the patriarch gave him a tenth of the spoils. 5 And those descendants of Levi who receive the priestly office have a commandment in the law to collect tithes from the people, that is, from their kindred, though these also are descended from Abraham. 6 But this man, who does not belong to their ancestry, collected tithes from Abraham and blessed him who had received the promises. 7 It is beyond dispute that the inferior is blessed by the superior. 8 In the one case, tithes are received by those who are mortal; in the other, by one of whom it is testified that he lives. 9 One might even say that Levi himself, who receives tithes, paid tithes through Abraham, 10 for he was still in the loins of his ancestor when Melchizedek met him.
The argument essentially is: Look! Genesis 14 displays Melchizedek as superior to Abraham, which also means he’s superior to Abraham’s descendant, Levi. So if Christ inherits Melchizedek’s priesthood, then his priesthood is de facto greater than the Levitical order.
11 Now if perfection had been attainable through the levitical priesthood—for the people received the law under this priesthood—what further need would there have been to speak of another priest arising according to the order of Melchizedek, rather than one according to the order of Aaron? 12 For when there is a change in the priesthood, there is necessarily a change in the law as well. 13 Now the one of whom these things are spoken belonged to another tribe, from which no one has ever served at the altar. 14 For it is evident that our Lord was descended from Judah, and in connection with that tribe Moses said nothing about priests.
Building off of Psalm 110:4. Basically saying, “Why would God promise us a priest like Melchizedek if the Levites were enough? Answer: They weren’t enough! We needed a better priest. So stop trying to go back to the Levitical priestly system! The Old Covenant has been superseded!”
15 It is even more obvious when another priest arises, resembling Melchizedek, 16 one who has become a priest, not through a legal requirement concerning physical descent, but through the power of an indestructible life. 17 For it is attested of him,
“You are a priest forever,
according to the order of Melchizedek.”
And here finally the author of Hebrews slides in his key quote from Psalm 110 to clinch the argument. The Heir of David (Jesus) is a priest in the order/manner of Melchizedek. He has a legitimate claim to priesthood that is even more ancient than that of the tribe of Levi.
18 There is, on the one hand, the abrogation of an earlier commandment because it was weak and ineffectual 19 (for the law made nothing perfect); there is, on the other hand, the introduction of a better hope, through which we approach God.
20 This was confirmed with an oath; for others who became priests took their office without an oath, 21 but this one became a priest with an oath, because of the one who said to him,
“The Lord has sworn
and will not change his mind,
‘You are a priest forever’”—
22 accordingly Jesus has also become the guarantee of a better covenant.
Levitical priests were ordained according to bloodline, but Melchizedekian priests (i.e. David’s line) got their privileges by covenant from God (as enshrined in Psalm 110). The author of Hebrews uses this as a neat little conceptual linkage to his next subject, which is how Christ has brought about the New Covenant promised by the OT prophets. He’s the better High Priest and the Guarantor of the New Covenant, which Christians get to participate in. So why go back to the Levitical sacrificial system?
So that’s essentially the low-down on Melchizedek. He is a foreshadowing of Christ and he establishes a precedent for a non-Levite being a legitimate priest (and king!) in the eyes of the true God.
 Bruce K. Walke, “Melchizedek,” in The Zondervan Pictorial Encyclopedia of the Bible, Vol. 4: M-P (Grand Rapids: Zondervan, 1976), 177.
 John H. Walton, Victor H. Matthews, & Mark W. Chavalas, The IVP Bible Background Commentary: Old Testament (Downers Grove: InterVarsity Press, 2000), 553.
 Ibid., 47.
 Peter T. O’Brien, Hebrews, PNTC (Grand Rapids: Eerdmans, 2010), 249.