The Levitical Priesthood - Members of the tribe of Levi were commissioned as priests following the sin of the golden calf. These are the priests who sacrificed animals to cover the sins of the people of Israel. They did this in the wilderness, the Tent of Meeting, and later in the Temple or Beit Hamikdash. Much of Leviticus details instructions for this priesthood. Only men from Levi were permitted to serve as priests and be part of Temple services.
The Melchizedek Priesthood - This is the order that the Messiah is high priest of. Melchizedek is not mentioned by name a lot in Scripture, but the writer of Hebrews had "much to say" about this priesthood. In Psalm 110:4, YHWH is talking to the Messiah and says, "You are a priest forever, according to the order of Melchizedek." This is actually the most quoted verse of the Old Testament in the New Testament, so we know it's important! The Melchizedek priesthood was in operation during the time of Abraham (Genesis 14:18). Since Melchizedek means 'King of Righteousness', we can see that Peter was imploring us to be part of the royal or King priesthood in 1 Peter 2:19.
Key Differences Between the Melchizedek and Levitical Priesthoods:
-Perfection is found in the Melchizedek priesthood, but not in the priesthood of Levi. (Hebrews 7:11)
-Scripture considers Melchizedek greater than Levi. (Hebrews 7:7)
-The Melchizedek priesthood needs only the blood of the Messiah, and does not require animal sacrifice. (Hebrews 10:11-12)
-The Messiah did not sin and therefore did not need to atone for himself, whereas priests of Levi would first have to sacrifice sin offerings for themselves, before they could do so for the people. (Hebrews 7:27)
-Melchizedek is forever but the Levitical priesthood was only for a time. (Hebrews 9:10)
-The Levitical priesthood system is a copy and shadow of the heavenly Melchizedek priesthood. (Hebrews 8:5)
-It is impossible for the blood of bulls and goats (the operation of the Levitical priesthood) to take away sin, but the Messiah saves completely through his priesthood. (Hebrews 10:4, 7:25)
-Levitical sacrifices had to be offered up day by day, but the Messiah's offering of his blood under the Melchizedek priesthood was once and for all. (Hebrews 7:27)
It's important to note that many English translations of Scripture fail to make this distinction, and will often use the same terms for both. Many translations of New Testament passages use "law" when it is actually talking about the Levitical priesthood Temple system. Which has caused much confusion about the value of YHWH's instructions for our lives. Learning to discern what the author of Scripture actually means when you see words like priesthood, law, or even Torah is key to correctly understanding the word!