The concept of one having the power of the priesthood is generally misunderstood in the Church. There is a difference between having the authority of the priesthood and having power in the priesthood. One does not have to be a priesthood holder in order to have power in the priesthood.
Men who hold the priesthood have the authority to perform the ordinances of the priesthood. Holders of the priesthood also have priesthood-mandated responsibilities to serve others. For example, a Bishop has responsibilities of service that come from virtue of his calling for which those who don’t hold that calling are not accountable. This is also true for all the presiding priesthood quorums of the Church.
The presiding quorums of the Church (First Presidency, Quorum of the Twelve, and the Quorums of the Seventy) all have God-given expectations of types of service for which those not in those quorums are not accountable. The work of these priesthood quorums are special and are related to the fact that they are serving the whole family of God. All of these Brethren are under obligation to travel the world and represent the Lord in their service to the human family. They regulate the affairs of the Lord’s kingdom on earth, and exercise the keys of the priesthood in performing their responsibilities.
Definitions of priesthood
There is a wonderful article associated with Sunday School lesson #25 (Church History) on lds.org. You can read it here. I highly recommend you read it very carefully, as it broadens our understanding of what the priesthood is, and how priesthood blessings and power in the priesthood are available to both men and women alike. My whole life I have understood power in the priesthood to be something only available to those who held the priesthood authority. Many of us still view the priesthood as the world defines priesthood. I will be including a number of quotes from the article I linked to in this paragraph. Many of them are placed together here, but are in different parts of the original article.
In 1830s America, the word priesthood was defined as “the office or character of a priest” and “the order of men set apart for sacred offices,” identifying priesthood with religious office and the men who held it. Early Latter-day Saints likewise thought of priesthood primarily in terms of ordination to ecclesiastical office and authority to preach and perform religious rites. As in most other Christian denominations during this era, Latter-day Saint men alone held priesthood offices, served formal proselytizing missions, and performed ordinances like baptism and blessing the sacrament of the Lord’s Supper.
In other words, in the early years of the Church the understanding of what the priesthood was very closely mirrored the world’s definition of what the word priesthood meant. But Joseph Smith quickly began to expand that definition and broaden our understanding of what is included in the word priesthood.
When the Relief Society was organized, it was originally just to be a charitable institution.
As Sarah Granger Kimball, a founding member of the Relief Society, later recalled, the Prophet told them he had “something better” for them and said he would organize the women “in the Order of the Priesthood after the pattern of the Church.”
The women were organized, as Apostle John Taylor remarked at the founding meeting, “according to the law of Heaven.”
For the first time women were given delegated keys to operate under the direction of those holding the keys of the priesthood. They were organized under the direction of a presidency, just like a priesthood quorum, though none of the women actually held priesthood authority. The Young Women’s organization and the Primary are now organized in the same manner, using delegated priesthood authority to act as a presidency to direct the affairs of their respective organizations under the auspices of those who hold the priesthood keys.
Women did receive authority to preside in the women’s organization and to appoint officers as needed to conduct the organization in the pattern of the priesthood, including being led by a president with counselors.
Already the Church was treating priesthood differently than any other Christian entity had ever envisioned it to be. Not only were laymen being given the priesthood, without training or pay, but now women were operating almost like clergymen under the direction of those who held the keys of the priesthood authority.
Along came temples
The advent of temple ordinances and covenants further expanded how the latter-day saints viewed and used the word priesthood.
Joseph Smith said that his instructions to the Relief Society were intended to prepare women to “come in possession of the privileges & blessings & gifts of the priesthood.” This would be accomplished through the ordinances of the temple.
When Joseph organized the Relief Society he spoke of establishing a “kingdom of priests.” This kingdom was to be made up of both men and women who made temple covenants. “He taught men and women that by receiving temple ordinances, culminating in the sealing ordinance, they entered into an “order of the priesthood.””
Temple ordinances are priesthood ordinances, but they do not bestow any kind of priesthood office on either men or women. What these ordinances do is enable the fulfilling of the Lord’s promise that His people would be “endowed with power from on high.”
That priesthood power was manifest in individuals’ lives in many ways and was available to adult members, regardless of marital status. The endowment opened channels of personal revelation to both women and men. It bestowed a greater measure of “faith and knowledge” and the “help of the Spirit of the Lord”
It prepared endowed Latter-day Saints to go forth “armed with thy [God’s] power” to “bear exceedingly great and glorious tidings … unto the ends of the earth.” Indeed, through the ordinances of the temple, the power of godliness was manifest in their lives.
These revelations and ordinances imparted new understanding of the interdependent relationship of women and men. As Bishop Newel K. Whitney expressed it shortly after receiving his endowment, “Without the female all things cannot be restor’d to the earth. It takes all to restore the Priesthood.” Mary Isabella Horne, a member of the Nauvoo Relief Society, later expressed joy in being “co-laborers with our brethren in building up the kingdom of God.” “In all the ordinances received in the House of the Lord,” she said, “woman stands beside the man, both for the living and the dead, showing that the man is not without the woman nor the woman without the man in the Lord.”
“When a man and a woman are sealed in the temple, they enter together, by covenant, into an order of the priesthood.”
The priesthood authority exercised by Latter-day Saint women in the temple and elsewhere remains largely unrecognized by people outside the Church and is sometimes misunderstood or overlooked by those within. Latter-day Saints and others often mistakenly equate priesthood with religious office and the men who hold it, which obscures the broader Latter-day Saint concept of priesthood.
Priesthood blesses the lives of God’s children in innumerable ways. Priesthood defines, empowers, ennobles, and creates order. In ecclesiastical callings, temple ordinances, family relationships, and quiet, individual ministry, Latter-day Saint women and men go forward with priesthood power and authority. This interdependence of men and women in accomplishing God’s work through His power is central to the gospel of Jesus Christ restored through the Prophet Joseph Smith.
Looking at the big picture
Power in the priesthood can be had through a priesthood holder magnifying his calling in the priesthood so that the possible good he can do is enlarged through his service. All priesthood holders are commanded to magnify their callings. But power in the priesthood comes to all of God’s covenant children, regardless of gender, through obedience to the covenants they make in the temple.
The power of the priesthood comes from the covenants we make. When we are baptized we receive the gift of the Holy Ghost. Through this member of the Godhead we are instructed and lead through the trials of mortality. Through the temple covenants we make to be obedient to God’s commandments we find refuge in the Savior’s atoning sacrifice, receive forgiveness of sins, develop traits of godliness, and become a holy people. It is through the blessings of the priesthood that all of God’s children become “chosen” by letting go of the desire for worldly possessions and the praise of the world.
Because power in the priesthood is available equally to men and women alike, we see that truly God is no respecter of persons. All of the blessings and accomplishments possible by being a member of Christ’s Church come because of the use and blessings of His priesthood. Both men and women share those blessings equally. This is how the Lord is building a generation of Priests, a royal priesthood. It is all thanks to the blessings that come from keeping the covenants made by participating in priesthood ordinances administered through those who hold the keys of the priesthood, and through those who operate under the delegated authority of that priesthood in the temples of God.