Webster's Dictionary 
(1): ( n.) The office or character of a priest; the priestly function.
(2): ( n.) Priests, taken collectively; the order of men set apart for sacred offices; the order of priests.
King James Dictionary 
PRIE'STHOOD, n. The office or character of a priest.
1. The order of men set apart for sacred offices the order composed of priests.
International Standard Bible Encyclopedia 
1. Priesthood is an Office
2. In the Old Testament
3. Hereditary Priesthood
4. In the New Testament
All worship is based on priesthood, for the priestly office is an essential part of salvation. Christianity itself has its glorious Priest, the Lord Jesus Christ, and it is through His one supreme offering that we are brought into saved relations with God and enjoy fellowship with Him. The priesthood of Christ and its mighty effects in sacrifice and intercession on behalf of the people of God are the chief and fundamental theme of the Epistle to the Hebrews.
1. Priesthood an Office:
Priesthood is a real office, definite and specific. It is needful to insist on this fact, for the noble word "priest" has been misappropriated and misapplied, so that its intrinsic import has been impaired. There is a certain literary slang indulged in by some who talk of the "priests of science," "priests of art," and similar absurdities. The idea of priesthood, if priesthood is to have any definite meaning, can have no place in literature or science or art or in anything of the kind. For it belongs to the realm of grace, presupposing as it does sin and the divine purpose to remove it. Hugh Martin writes that he "would as soon think of transferring the language of geometry and of algebra to botany and talk of the hypothenuse of a flower and the square root of a tree, or the differential coefficient of a convolvulus, as to speak of the priesthood of nature or letters." Priesthood is an office, embracing very specific duties and functions.
2. In the Old Testament:
Priesthood in some form appears to have existed from the earliest times, even from the beginning of the history of our race. In patriarchal times the office was held and its duties were discharged by those who occupied some sort of headship, and particularly by the father or the chief of the family and of the tribe. Thus, Noah in his capacity of priest and in behalf of his household "builded an altar unto Yahweh, and took of every clean beast, and of every clean bird, and offered burnt-offerings on the altar" ( Genesis 8:20 ). Abraham offered the ram "for a burnt-offering in the stead of his son" ( Genesis 22:13 ). In like manner Job offered burnt offerings for his children, and likewise by divine direction for the three "comforters" when the great trial had passed ( Job 1:5; Job 42:8 ). In these and the like instances there was priestly action no less certainly than in that of Aaron or of any regularly appointed priest in Israel. Melchizedek was "priest of God Most High" ( Genesis 14:18 ). Isaac "builded an altar there and called upon the name of Yahweh" ( Genesis 26:25 ), as did Jacob ( Genesis 33:20 ). In these cases priestly acts were performed by the patriarchs in their capacity as fathers of the family or heads of clans. From the beginning, priesthood with its acts of expiation and of worship was thus recognized as a divinely-instituted office. But in pre-Mosaic times there was no special class of priests recognized.
3. Hereditary Priesthood:
Regular priestly succession in a single family was established by Moses ( Exodus 28:1-3 ). From this point of time onward the priesthood in Israel was confined to the family of Aaron. No hereditary priesthood seems to have prevailed in patriarchal times. According to the Epistle to the Hebrews, Melchizedek, a priest of the highest rank, had neither predecessor nor successor in his great office. By divine direction Moses designated the Aaronic family as the priestly family in Israel, and he prescribed the garments they should wear, the sacrifices they should offer both for themselves and for the congregation, their maintenance, their domestic relations, and their conduct toward their fellow Hebrews.
In the appointment of the priesthood there is no trace of Egyptian influence. Yet we know that Joseph married the daughter of the priest of On ( Genesis 41:50 ). But this fact had no bearing on the selection of Israel's priestly family. The Aaronic priesthood had nothing in common with that of Egypt; it claimed to be of divine origin, and its duties, functions and powers in no way contradict the claim. The witness of an Egyptian archaeologist (Dr. M.G. Kyle) may be here introduced touching one essential element in the duties of the priestly office, namely, sacrifice: "The entire absence from the offerings of old Egyptian religion of any of the great Pentateuchal ideas of sacrifice, substitution, atonement, dedication, fellowship, and indeed of almost every essential idea of real sacrifice, as clearly established by recent very exhaustive examination of the offering scenes, makes for the element of revelation in the Mosaic system by delimiting the field of rationalistic speculation on the Egyptian side. Egypt gave nothing to that system, for it had nothing to give." As much may be said respecting the priesthood; Israel took little or nothing of its powers and functions from Egyptian sources.
Although the office was limited to the Aaronic family, nevertheless in certain exigencies and emergencies others beside the regular priest offered sacrifices to the Lord and were accepted by Him. Thus did Gideon in a time of great straits in Israel ( Judges 6:24 , Judges 6:26 ); thus the men of Beth-shemesh ( 1 Samuel 6:14 , 1 Samuel 6:15 ); the prophet Samuel ( 1 Samuel 7:9 ); David ( 2 Samuel 6:13 , 2 Samuel 6:17 ); Elijah ( 1 Kings 18:23 , 1 Kings 18:32-38 ), etc. The chosen people appear to have felt free to offer sacrifices and to engage in priestly functions when occasion required, until the central sanctuary was established on Mt. Moriah. When the Temple was built and dedicated, priestly action was confined to Jerusalem and to the regular priestly household. When Pharisaism, with its rigid legalism, with its intolerable burdens, became dominant, all liberty of worship and spontaneous service largely disappeared. The religious life of Israel stiffened into a dreadful monotony.
4. In the New Testament:
All priesthood reaches its climax in that of the Lord Jesus Christ. It is because of the perfection of His priesthood that the office as represented by Melchizedek and Aaron was effective, and fulfilled the end for which it was appointed. The one answers to the other as type and antitype, as prediction and fulfillment. Christ's priesthood is opened to us in the Epistle to the Hebrews ( Matthew 2:14-18; Matthew 4:14-16; Matthew 5:1-10; Matthew 7:9 , Matthew 7:10 , Matthew 7:18 ). Two fundamental truths touching His priesthood are made very prominent in the Epistle to the Hebrews. These are its order and its duties. By the order is meant the rank or grade of the Priest, and by the duties the various functions of His ministry. Christ's order as Priest is that of Melchizedek, not at all that of Aaron; Heb 7 makes this fact perfectly clear. Like Melchizedek, and infinitely above Melchizedek, He is Priest, having no predecessor in the great office, and no successor; herein He stands absolutely alone, peerless and perfect forever. He executes the duties or functions of it after the pattern of Aaron, as Heb 9 clearly exhibits. These two priesthoods, Melchizedek's and Aaron's, are gloriously accomplished in the person and Work of Jesus Christ.
The point is raised and discussed with some keenness in our day, Did Christ execute the office of priest during His sojourn on earth, or does He exercise the office only in heaven? A full discussion of this interesting subject would be inappropriate. However, let it be noted (1) that the Lord Jesus was appointed a Priest no less certainly than was Aaron ( Hebrews 5:4 , Hebrews 5:5 ). In the words, "Thou art my Son, this day have I begotten thee," there appears to be a reference both to His incarnation ( Luke 1:32; Hebrews 1:5 ) and also to His resurrection ( Acts 13:33 ). In Hebrews 2:17 we are told that it "behooved him in all things to be made like unto his brethren, that he might become a merciful and faithful high priest in things pertaining to God, to make propitiation for the sins of the people." The assumption of human nature was needful that He might be such a priest. John the Baptist saw this truth, and said, "Behold the Lamb of God, that taketh away the sin of the world" ( John 1:29 ).
There was certainly priestly action in His death. Twice we are told that He "offered up himself" ( Hebrews 7:27 ), "For this he did once for all, when he offered up himself." This strong term, "offered," is sacrificial and points to His death as an offering made for the sins of the people. His own action in it must not be overlooked; it was He Himself who presented the offering; He was not, therefore, a struggling victim, a martyr, who could not escape the doom that came upon Him - nay, He voluntarily offered Himself.
In Hebrews 9:14 we find these significant words: "How much more shall the blood of Christ, who through the eternal Spirit offered himself without blemish unto God, cleanse your conscience from dead works to serve the living God?" It was as Priest that He made this stupendous offering, and this He did when still on earth. He was at once both sacrifice and priest. Never was He more active than when He offered Himself to God.
It is worthwhile to remind ourselves that the words employed in Scripture to express the act of His dying are never used to denote the death of a creature, a man. Matthew has, He "yielded up (dismissed), his spirit" ( Matthew 27:50 ). John has, He "gave up his spirit" ( John 19:30 ); Mark 15:37 and Luke 23:46 both have the same words: He "gave up the ghost." He died, not because He was mortal as we are, nor because He could not deliver Himself, but because He gave Himself for our sins that we might be forgiven and saved ( John 10:17 , John 10:18 ). The voluntariness of His offering is the very essence of His priestly atonement. See Christ , Offices Of , V; Priesthood In The New Testament .
Priesthood springs out of the deepest need of the human soul. Men universally feel that somehow they have offended the Power to whom they are responsible, to whom they must give account of their deeds. They long to appease their offended Lord, and they believe that one who is authorized and qualified to act in their behalf may secure for them the abrogation of penalty and the pardon they seek. Hence, priesthood connects itself most closely with sin, with guilt and its removal. The heart craves the intervention and intercession on their behalf of one who has liberty of access to God, and whose ministry is acceptable. In short, the priest is the representative of the sinner in things pertaining to God. He is the mediator whose office it is to meet and satisfy the claims of God upon those for whom he acts, and who secures the pardon and the favor which the offender must have, if he is to enjoy fellowship with God. And this, and more than this, we have in our Great High Priest, the Lord Jesus Christ.