In the Mormon church, all power flows through the all-male priesthood which consists of all Mormon men above a certain age. Priests hold all of the religious authority for sealing, binding, and loosing — the central sacred acts of the Mormon Church which serve to determine every Mormon's eternal destiny.
These priests hold all of the keys to eternal life and are themselves the key to the claim of the Church of Jesus Christ of Latter-Day Saints that it is the only legitimate Christian church on earth.
Source of Sacred Power in the Mormon Church
A vital question in any church is: who holds sacred power and why? Where does authority flow from and how? In Christianity, ultimate authority rests with God and Jesus Christ, but how do mere mortal humans come by any authority to speak for God and instruct other humans about what they should or should not do?
Every Christian movement has developed different ways to answer this question of the source and nature of religious authority. Joseph Smith, because he rejected all past and present Christian churches as corrupt abominations, could not use them as models or rely upon them.
Fortunately for Smith, John the Baptist paid him a visit and conferred directly upon him priestly authority as well as the authority to create new priests. John the Baptist lived and worked before Christianity and Christian chuches existed. This means that the authority he conferred on Joseph Smith was not only independent of all the "corrupt" churches which appeared after the death of Jesus, but ultimately derived from the traditions of ancient Israel rather than from ancient Christian churches (like Roman Catholicism, for example).
The Jewish roots of Joseph Smith's ideas about priesthood are demonstrated in the names given to the two levels of Mormon priesthood: Aaronic and Melchizedek. In this way, centuries of Christian traditions, structures, and beliefs were swept aside in favor of something completely new which was based, at least in theory, on something much older.
Aaronic and Melchizedek Priesthoods
The first and most basic level of priesthood in the Mormon Church is the Aaronic priesthood. This is open to every "worthy" Mormon male at the age of 12 and consists of most Mormon males between 12 and 17. The purpose of the Aaronic priesthood is basically to prepare young Mormon men for the leadership responsibilities of the Melchizedek priesthood. Tasks include blessing and serving communion, teaching, baptizing, and ordaining other boys into the Aaronic priesthood.
The Melchizedek priesthood is the core of the Momron church's power structure. It is named after the high priest Melchizedek, described in the Old Testament as having lived at the time of Abraham. Priests at this do many of the same things as those in the Aaronic priesthood, plus a variety a administrative tasks, as well as more serious sacraments like marriage.
A couple of important features distinguish the two, though. Only Melchizedek priests can live eternally in the Celestial Kingdom, the highest of the three heavens in Mormon teachings. Only Melchizedek priests can receive a temple endowment and become "sealed" to a wife for all eternity. Finally, Melchizedek priests are the only source of salvation for any human being.
According to Bruce McConkie:
"Without the Melchizedek Priesthood salvation in the kingdom of God would not be available for men on earth. ... As far as all religious organizations now existing are concerned, the presence or the absence of this priesthood establishes the divinity or falsity of a professing church."
Sacred Power and Function of Mormon Priests
The Mormon Church does not have a "professional" class of priests — people whose job it is to be priests and nothing else. In the Mormon church, the power and authority of the priesthood is spread out among just about all male Mormons.
This means that just about every male Mormon has the sort of religious, sacred power and authority that one finds among Catholic priests. Bishops in the Mormon church, even if they are also politicians like Mitt Romney, have the same religious and sacred status in the Mormon church as Catholic or Anglican bishops.
At one time this sacred power and authority was denied to black men. A new revelation from God was required for Mormons to decide that black men should be treated as equals to white men. The fact that earlier black men must have been excluded from the highest Mormon heaven isn't something Mormons like to dwell on.
Women continue to be excluded from the sacred power and authority of Mormon priesthood. This isn't unusual among Christian churches, but given how much more widely religious authority is distributed in the Mormon church, the exclusion of all women is both more obvious and, arguably, more serious. Other churches may restrict the priesthood to men, but they also restrict it to just some men.