What laws and priesthood do we currently live under?
Key Verse:  As he saith also in another place, Thou art a priest for ever after the order of Melchisedec.  [Hebrews 5:6]
In the book of Hebrews Paul writes:

So also Christ glorified not himself to be made an high priest; but he that said unto him, Thou art my Son, to day have I begotten thee.  As he saith also in another place, Thou art a priest for ever after the order of Melchisedec.  [Hebrews 5:5-6]

So what does this mean, that Christ is a high priest forever after the order of Melchisedec?  To find the answer we have to go back to the beginning.

Creation

At the creation God established a basic set of laws for the benefit of man.  Many references to these laws exist in the book of Genesis, including:

Seventh Day for Rest

And on the seventh day God ended his work which he had made; and he rested on the seventh day from all his work which he had made.  And God blessed the seventh day, and sanctified it: because that in it he had rested from all his work which God created and made.  [Genesis 2:3]

Thou shall not kill

And Cain talked with Abel his brother: and it came to pass, when they were in the field, that Cain rose up against Abel his brother, and slew him.  And the LORD said unto Cain, Where is Abel thy brother? And he said, I know not: Am I my brother's keeper?  And he said, What hast thou done? the voice of thy brother's blood crieth unto me from the ground.  And now art thou cursed from the earth, which hath opened her mouth to receive thy brother's blood from thy hand;  When thou tillest the ground, it shall not henceforth yield unto thee her strength; a fugitive and a vagabond shalt thou be in the earth.  [Genesis 4:8-12]

Law of the Clean and  Unclean

Of every clean beast thou shalt take to thee by sevens, the male and his female: and of beasts that are not clean by two, the male and his female.  [Genesis 7:2]

Thou shall not commit adultry

And it came to pass, that, when Abram was come into Egypt, the Egyptians beheld the woman that she was very fair.  The princes also of Pharaoh saw her, and commended her before Pharaoh: and the woman was taken into Pharaoh's house.  And he entreated Abram well for her sake: and he had sheep, and oxen, and he asses, and menservants, and maidservants, and she asses, and camels.  And the LORD plagued Pharaoh and his house with great plagues because of Sarai Abram's wife.  And Pharaoh called Abram, and said, What is this that thou hast done unto me? why didst thou not tell me that she was thy wife?  [Genesis 12:14-18]

Tithing

And Melchizedek king of Salem brought forth bread and wine: and he was the priest of the most high God.  And he blessed him, and said, Blessed be Abram of the most high God, possessor of heaven and earth:  And blessed be the most high God, which hath delivered thine enemies into thy hand. And he gave him tithes of all.  [Genesis 14:18-20]

For this Melchisedec, king of Salem, priest of the most high God, who met Abraham returning from the slaughter of the kings, and blessed him;  To whom also Abraham gave a tenth part of all; first being by interpretation King of righteousness, and after that also King of Salem, which is, King of peace;  [Hebrews 7:1-2]

After the Exodus from Egypt, the children of Israel were given the Ten Commandments on two tables of stone, written by the finger of God and given to Moses.  These tables of stone were put inside the ark of the covenant.

And he gave unto Moses, when he had made an end of communing with him upon mount Sinai, two tables of testimony, tables of stone, written with the finger of God.  [Exodus 31:18]

There was nothing in the ark save the two tables of stone, which Moses put there at Horeb, when the LORD made a covenant with the children of Israel, when they came out of the land of Egypt.  [1 Kings 8:9]

Exodus chapter 20 contains the text of these commandments.

Levitical Priesthood

While Moses was receiving the Ten Commandments on mount Sinai, the children of Israel were building a molten calf out of gold and proclaiming it to be the god who had brought them out of Egypt.  When Moses came down from the mount, he was extremely angry.  He smashed the tables of stone containing the Ten Commandments, ground up the golden calf, put the dust on the water, and made the children of Israel drink it.  Then Moses asked the question "Who is on the LORD'S side?"

Then Moses stood in the gate of the camp, and said, Who is on the LORD'S side? let him come unto me. And all the sons of Levi gathered themselves together unto him.  [Exodus 32:26]

At that time the LORD separated the tribe of Levi, to bear the ark of the covenant of the LORD, to stand before the LORD to minister unto him, and to bless in his name, unto this day.  Wherefore Levi hath no part nor inheritance with his brethren; the LORD is his inheritance, according as the LORD thy God promised him.  [Deuteronomy 10:8-9]

The sons of Levi answered the call and God made them priests over the children of Israel.  Because the children of Israel committed this transgression with the golden calf, the Levitical priesthood was established and the added law was given to the children of Israel.

And it came to pass, when Moses had made an end of writing the words of this law in a book, until they were finished,  That Moses commanded the Levites, which bare the ark of the covenant of the LORD, saying,  Take this book of the law, and put it in the side of the ark of the covenant of the LORD your God, that it may be there for a witness against thee.  For I know thy rebellion, and thy stiff neck: behold, while I am yet alive with you this day, ye have been rebellious against the LORD; and how much more after my death?  [Deuteronomy 31:24-27]

Wherefore then serveth the law? It was added because of transgressions, till the seed should come to whom the promise was made; and it was ordained by angels in the hand of a mediator.  [Galatians 3:19]

Note three points about the two laws here.  1) The added laws was written by the hand of Moses while the ten commandments were written by the finger of God.  2) The book of the law was placed in the side of the ark while the ten commandments on the two tables of stone were placed in the ark.  3) The added law was given as a witness against the children of Israel and was added because of trangression.  A clear distinction was made between the two laws.

Melchisedec Priesthood

The Levitical priesthood lasted from the time of Sinai until Jesus came to the earth.  At this time Jesus was installed as our high priest after the order of Melchisedec.

For he testifieth, Thou art a priest for ever after the order of Melchisedec.  [Hebrews 7:17]

If therefore perfection were by the Levitical priesthood, (for under it the people received the law,) what further need was there that another priest should rise after the order of Melchisedec, and not be called after the order of Aaron?  For the priesthood being changed, there is made of necessity a change also of the law.  [Hebrews 7:11-12]

This last verse makes reference to a change in the law.  So what law was changed by Christ?  Paul gives use the answer in writings to three different churches.

And you, being dead in your sins and the uncircumcision of your flesh, hath he quickened together with him, having forgiven you all trespasses;  Blotting out the handwriting of ordinances that was against us, which was contrary to us, and took it out of the way, nailing it to his cross;  [Colossians 2:13-14}

For he is our peace, who hath made both one, and hath broken down the middle wall of partition between us;  Having abolished in his flesh the enmity, even the law of commandments contained in ordinances; for to make in himself of twain one new man, so making peace;  And that he might reconcile both unto God in one body by the cross, having slain the enmity thereby:  [Ephesians 2:14-16]

For as many as are of the works of the law are under the curse: for it is written, Cursed is every one that continueth not in all things which are written in the book of the law to do them.  But that no man is justified by the law in the sight of God, it is evident: for, The just shall live by faith.  And the law is not of faith: but, The man that doeth them shall live in them.  Christ hath redeemed us from the curse of the law, being made a curse for us: for it is written, Cursed is every one that hangeth on a tree:  That the blessing of Abraham might come on the Gentiles through Jesus Christ; that we might receive the promise of the Spirit through faith.  [Galatians 3:10-14]

These three sets of verses state that Christ "blotted out the handwriting of ordinances", "abolished . . . the law of commandments contained in ordinances", and "redeemed us from the curse of the [book of the] law".  The added law was the only law that was written by the hand of man; the ten commandments were written by the finger of God.  Also from Deuteronomy 31:26 we know that the added law was "a witness against thee".  It is clear that the reference to "the handwriting of ordinances that was against us" is a reference to the added law, not the Ten Commandments.  If  "the handwriting of ordinances" refers to the added law, "the law of commandments contained in ordinances" must also refer to that law as they both refer to ordinances.  As "the book of the law" is the same name used for the added law in Deuteronomy, all three of these texts must refer to the added law given under the Levitical priesthood.   When Christ died for our sins, these added laws were done away with.

So now that Christ has redeemed us from the curse of the law, what statutes do we live under?  Since Christ is our high priest, we are now back under order of Melchisedec, or the Melchisedec priesthood.  We are under the laws of the Melchisedec priesthood.

Confusion often arises in the fact that many of the laws under the Melchisedec priesthood were repeated in the added laws of the Levitical priesthood.  The fact that the added laws were done away with does not mean that equivalent laws under the Melchisedec priesthood were also done away with.  A good example of this is the law against murder, which is contained in both the Ten Commandments and the added law.

Thou shalt not kill.  [Exodus 20:13]

And he that killeth any man shall surely be put to death.  [Leviticus 24:17]

Just because the Levitical law against murder was blotted out doesn't mean that murder is not now a sin.  Jesus is our high priest after the order of Melchisedec and the laws of the Melchisedec priesthood, referenced in the books of Genesis and Exodus before the establishment of the Levitical priesthood, are still in force.

Two other statutes were added to the Melchisedec priesthood when Jesus became our high priest; the Lord's Supper and Baptism.

And he took bread, and gave thanks, and brake it, and gave unto them, saying, This is my body which is given for you: this do in remembrance of me.  Likewise also the cup after supper, saying, This cup is the new testament in my blood, which is shed for you.  [Luke 22:19-20]

Now when they heard this, they were pricked in their heart, and said unto Peter and to the rest of the apostles, Men and brethren, what shall we do?  Then Peter said unto them, Repent, and be baptized every one of you in the name of Jesus Christ for the remission of sins, and ye shall receive the gift of the Holy Ghost.  [Acts 2:38]

These two statutes are part of the testimony of Jesus.  We are required to keep this testimony and to model our lives after his example.

For even hereunto were ye called: because Christ also suffered for us, leaving us an example, that ye should follow his steps:  Who did no sin, neither was guile found in his mouth:  Who, when he was reviled, reviled not again; when he suffered, he threatened not; but committed himself to him that judgeth righteously:  [1 Peter 2:21-23]
For more information on the Melchisedec and Levitical laws, read the tract Does the Bible Contradict Itself - The Two Laws Contrasted.
Conclusion

Therefore, the law and commandments from God that we are to observe today, being under the Melchisedec priesthood and having Jesus Christ as our high priest, are: