While the responsibilities of priesthood under the Mosaic Law were clearly outlined in the Old Testament, many Christians don't realize that Jesus defined His priesthood in the New Testament.

Since Jesus said he 'would build his church,' shouldn't Jesus church today be setup today, with the same Priesthood that He established in His own day?  See the list  of Jesus' priesthood offices below to see and know that it still exists.

Purpose

What is the purpose of 'Priesthood?'   Priesthood carry out two specific purposes:  

1) To stand in the stead of Jesus to perform the ordinances of His Church ( Baptism, Communion, Laying on of Hands, etc.,)

2) To 'perfect' or teach people about God's laws, covenants, promises and plan of salvation.

History

Jesus' complete Priesthood historically started with Adam, and was passed down through to Abraham's sons, as explained in latter-day revelation:

DC 104:28a Three years previous to the death of Adam, he called Seth, Enos, Cainan, Mahalaleel, Jared, Enoch, and Methuselah, who were all high priests, with the residue of his posterity, who were righteous, into the valley of Adam-ondi-Ahman, and there bestowed upon them his last blessing.

While the High Priesthood existed prior to Moses, In Moses day, iniquity among God's covenant people caused some of the higher priesthood roles to be unfulfilled, for a time, in Israel:

Exodus 34:1 And the Lord said unto Moses, Hew thee two other tables of stone, like unto the first, and I will write upon them also, the words of the law, according as they were written at the first on the tables which thou brakest; but it shall not be according to the first, for I will take away the priesthood out of their midst; therefore my holy order, and the ordinances thereof, shall not go before them; for my presence shall not go up in their midst, lest I destroy them.  

The Nephites in America had the priesthood also, but their righteousness caused them to have the higher offices of priesthood restored even before Jesus' birth.

Why is Jesus Priesthood named after Melchisedec?

Jesus' priesthood has two divisions on earth: the Melchisedec, and the Aaronic, including the Levitical priesthood.

The real name of the priesthood is after Jesus name, since he is the author and ruler over it.  Melchisedec was a man who lived between Noah and Abraham. The reason why Melchisedec is mentioned is as follows:

DC 104:1b Why the first is called the Melchisedec priesthood, is because Melchisedec was such a great high priest: before his day it was called the holy priesthood, after the order of the Son of God;
1c but out of respect or reverence to the name of the Supreme Being, to avoid the too frequent repetition of his name, they, the church, in ancient days, called that priesthood after Melchisedec, or the Melchisedec priesthood.

So, to avoid using the Creator's name too casually, the priesthood title was given Melchisedec's name.  The real name for the priesthood is the 'Order of the Son of God.' (Genesis 14:28 , Hebrews 7:3 , Alma 9:73)

Moreover, the lesser priesthood, the 'Aaronic,' is designed to be an appendage of the higher, the Melchisedec: 

DC 104:8a The second priesthood is called the priesthood of Aaron, because it was conferred upon Aaron and his seed, throughout all their generations.
8b Why it is called the lesser priesthood is, because it is an appendage to the greater, or the Melchisedec priesthood, and has power in administering outward ordinances.

Didn't the Priesthood end with the Law of Moses?

Jesus priesthood existed prior to the law of Moses.  Some Christian's teach (mistakenly) that priesthood did not continue after the sacrifices, burnt offerings, etc., ended.  This is not true, as the list below, demonstrates:  ordinations to many offices of Jesus' priesthood continued with the Apostles (Mark 3:13, Acts 14:23 , I Timothy 2:7 , II Timothy 4:22 , Titus 1:5 , Titus 3:15 , Hebrews 5:1 , Hebrews 7:3 ).  

The purpose and power of this priesthood has been restored today.

Note that was a spiritual parallel to the priesthood under Moses exists:  While priests in Moses' day were asked to carry out many physical sacrifices and unusual (to us) offerings, they all had a spiritual lesson to teach.  While the physical offerings are done away, however, even Paul taught that the priesthood were to offer up 'spiritual sacrifices' unto God:

I Peter 2:5 Ye also, as lively stones, are built up a spiritual house, an holy priesthood, to offer up spiritual sacrifices, acceptable to God by Jesus Christ.

'Spiritual sacrifice' means especially to cleanse oneself from the sin of the world.  Priesthood are called to teach the people to do likewise, coming unto Jesus with broken hearts and contrite spirits:

Psalms 51:17 The sacrifices of God are a broken spirit; a broken and a contrite heart,

2 Nephi 1:72 Behold, he offereth himself a sacrifice for sin, to answer the ends of the law, unto all those who have a broken heart and a contrite spirit; and unto none else can the ends of the law be answered.

According to scripture, Jesus priesthood, established in the beginning of the world, shall exist in the end of the world also (Genesis 6:7 ).

The old testament documents that the Levites (i.e. male descendants of the tribe of Levi) were to be the priests over Israel:

Deuteronomy 21:5 And the priests the sons of Levi shall come near; for them the Lord thy God hath chosen to minister unto him, and to bless in the name of the Lord; and by their word shall every controversy and every stroke be tried;

Some confuse this issue and believe that no other man (of another tribe, or Gentile, for that matter) could be spiritually called to priesthood.  Not so.  If one considers that Jesus, himself, bore the title 'High Priest' of his priesthood (Hebrews 3:1 , Hebrews 4:14 , Hebrews 6:20 ), and realizes the Jesus, himself, was not of Levi, but of Judah, (Paul was of the tribe of Benjamin, Acts 13:21 , Romans 11:1  and was ordained an apostle, etc.) one realizes that Jesus' priesthood was not solely confined to the Levitical lineage.

Can anyone just 'claim' priesthood?  Who can have it?  

Two groups of people have claim on priesthood: those with a literal heritage from the House of Israel, and those spiritually called by God to have this responsibility.

First, literal descendants of the Levites of the House of Israel have claim to priesthood by lineage.  This is not only supported by modern revelation through Joseph Smith Jr. in 1830, but recent genetic studies have proven that a gene on the male chromosome exists of literal Levitical priesthood holders.  This genetic 'marker' is only found of those whose priesthood passed from father to son.  People of this order are found among orthodox Jews, and even people (correctly) claiming lineage through the lost tribes of Israel.  (Click here to read more on this current scientific discovery).  Food for thought:  how did Joseph Smith 'know' this would be a genetic truth, that literal descendants of Aaron could 'prove' their lineage, unless God truly revealed it to him?  In 1830, no such genetic evidence existed!

Second, those called by God.  Paul stated that no one could just 'sign up' for priesthood responsibility, or worse, pay to have it.

Hebrews 5:4 And no man taketh this honor unto himself, but he that is called of God, as was Aaron. (...and how was Aaron called?  by a prophet, Moses.)

In Paul's day, a man named Simon saw the apostles bestowing the Gift of the Holy Ghost by the laying on of hands.  Seeing this, he wanted to be able to do what they did because of their priesthood authority.  He offered money for the privilege.  (Bad move on his part.)  The apostles' response:

Acts 8:20 But Peter said unto him, Thy money perish with thee, because thou hast thought that the gift of God may be purchased with money.

Simon couldn't buy it.  Neither can anyone today 'buy' priesthood--it comes by the calling of God through his senior priesthood, by revelation only.

Hebrews 7:23 But this man, because he continueth ever, hath an unchangeable priesthood.

How was Priesthood 'handed down?'

DC 68:2g but by virtue of the decree concerning their right of the priesthood descending from father to son, they may claim their anointing, if at any time they can prove their lineage, or do ascertain it by revelation from the Lord under the hands of the above-named presidency.

DC 104:18 The order of this priesthood was confirmed to be handed down from father to son, and rightly belongs to the literal descendants of the chosen seed, to whom the promises were made. This order was instituted in the days of Adam, and came down by lineage in the following manner:

Also, interesting to note, that Joseph's words " ..rightly belongs to the literal descendants of the chosen seed..." have been genetically proven. Research in the last couple years has identified in the Cohen (Literal descendants of the tribe of Lehi--the orthodox Jewish priesthood) have found that a gene exists only within those who have priesthood.  It is only transferred from father to son.  It is on the Y-Chromosome.  Only men have Y-chromosomes; women do not. 

A Nova (PBS) show made this statement, almost identical to Joseph Smith's revelation nearly 160 years earlier:

"...The results are stunning. A group of genetic markers ...- is seen in about 10% of the general Jewish population and in over 50% of the Cohanim. The long-held Jewish belief that priestly status is passed from father to son over the centuries seems to be confirmed in the genes. " from http://www.pbs.org/wgbh/nova/transcripts/2706israel.html

 

Other related scientific articles:

http://www.ramsdale.org/dna13.htm

http://www.sciencenews.org/archives/19981003/bob1.asp

http://www.sciencedaily.com/releases/1998/07/980714071409.htm

 

Isn't it only the Inspired Version that talks about Melchisedec's priesthood?

No.  Recent evidence from the Dead Sea Scrolls have verified that the understanding given to  Joseph Smith regarding Melchisedec's priesthood was correct:  a dead sea scroll fragment passage calls him "a priest in the assembly of God":

[MLKY] CDQ KWHN B({*D}[T )L]

. . . Melchi]zedek, priest in the asse[mbly of God . . .

(From http://www.st-and.ac.uk/~www_sd/med_melchizedek_art.html )

Apparently God was 100% correct when he had Hebrews 7:3 corrected to read:

Hebrews 7:3 For this Melchizedek was ordained a priest after the order of the Son of God, which order was without father, without mother, without descent, having neither beginning of days, nor end of life. And all those who are ordained unto this priesthood are made like unto the Son of God, abiding a priest continually.

Having examined some of the history and foundation for the Scriptural case for priesthood, the specific offices of Jesus priesthood are listed as follows:

I Corinthians 12:28 And God hath set some in the church, first apostles,...

Acts 1:26 And they gave forth their lots; and the lot fell upon Matthias; and he was numbered with the eleven apostles.

DC 104:11

I Timothy 3:1-4

DC 129:8a

Hebrews 5:1-2

DC 120:2-3

II Timothy 4:5

4:5 But watch thou in all things, endure afflictions, do the work of an evangelist, make full proof of thy ministry.

DC 125:3-4

125:3a The patriarch is an evangelical minister. The duties of this office are to be an evangelical minister; to preach, teach, expound, exhort, to be a revivalist, and to visit branches and districts as wisdom may direct, invitation, request, or the Spirit of God determine and require; 125:3b to comfort the Saints; to be a father to the church; to give counsel and advice to individuals who may seek for such; to lay on hands for the conferment of spiritual blessing, and if so led, to point out the lineage of the one who is blessed. 125:4a He is to be free from responsibility--ministerial--as a traveling minister, and from the care of the local branch or church and district affairs.

Luke 10:1-2

10:1 After these things the Lord appointed other seventy also, and sent them two and two before his face, into every city and place where he himself would come.

DC 104:13

104:13a The Seventy are to act in the name of the Lord, under the direction of the Twelve, or the traveling high council, in building up the church, and regulating all the affairs of the same, in all nations; first unto the Gentiles, and then to the Jews;

Acts 14:23

14:23 And when they had ordained them elders in every church, and had prayed with fasting, they commended them to the Lord, on whom they believed.

DC 17:8-9

17:8a The duty of the elders, .... it is his calling to baptize, and to ordain other elders, priests, teachers, and deacons, and to administer bread and wine--the emblems of the flesh and blood of Christ-- 17:8c and to confirm those who are baptized into the church, by the laying on of hands for the baptism of fire and the Holy Ghost, according to the Scriptures; 17:8d and to teach, expound, exhort, baptize, and watch over the church; 17:8e and to confirm the church by the laying on of the hands, and the giving of the Holy Ghost, 17:8f and to take the lead of all meetings. 17:9 The elders are to conduct the meetings as they are led by the Holy Ghost, according to the commandments and revelations of God.

Priests:  Home Ministers, Teachers, Preachers...

Acts 6:6-7

6:6 Whom they set before the apostles; and when they had prayed, they laid their hands on them. 6:7 And the word of God increased; and the number of the disciples multiplied in Jerusalem greatly; and a great company of the priests were obedient to the faith.

DC 17:10

17:10a The priest's duty is to preach, teach, expound, exhort, and baptize, and administer the sacrament. 17:10b and visit the house of each member, and exhort them to pray vocally and in secret, and attend to all family duties: 17:10c and he may also ordain other priests, teachers, and deacons; 17:10d and he is to take the lead of meetings when there is no elder present, but when there is an elder present he is only to preach, teach, expound, exhort, and baptize, and visit the house of each member, exhorting them to pray vocally and in secret, and attend to all family duties. 17:10e In all these duties the priest is to assist the elder if occasion requires.

Teachers:  Peacemakers, "Aaronic Patriarchs"...

Acts 13:1-2

13:1 Now there were in the church that was at Antioch certain prophets and teachers; as Barnabas, and Simeon that was called Niger, and Lucius of Cyrene, and Manaen, which had been brought up with Herod the tetrarch, and Saul. 13:2 As they ministered to the Lord,...

DC 17:11

Deacons:  Temporal Ministry for the Local Church

I Timothy 3:8-10

3:8 Likewise the deacons must be grave, not double-tongued, not given to much wine, not greedy of filthy lucre; 3:9 Holding the mystery of the faith in a pure conscience. 3:10 And let these also first be proved; then let them use the office of a deacon, being found blameless.

DC 83:22-23