That power is real, but it doesn't fit into my late 20th- and early 21st-century world. Nor is it something that I can control like a comic book superhero. It is not something I can will, though I can make it less likely that God's power will manifest itself through me by denying my relationship with and dependence on him through sin.
There is nothing to which I can compare priesthood power. It isn't magic or occult, but it also isn't something that empirical science can investigate. We don't have a category available into which I can put it, so I am unable to say what it is without simply restating that it is the power of God delegated to human beings.
But thinking about the power of the priesthood makes me think there is a better way to talk about it: it isn't so much that the person ordained to the priesthood receives power, as that he is designated as a person through whom the power of God can work. Delegation of authority and priesthood power go hand-in-hand.
The founding prophet of the LDS Church, Joseph Smith, taught that priesthood power is eternal, existing "with God from eternity, and . . . to eternity" (Alma P. Burton, Discourses of the Prophet Joseph Smith, 48). It makes sense to say that God's power is co-eternal with him. Given that, we can revise the standard formula: to a Mormon the power of the priesthood is not only the power to act in the name of God, it is also the power of God acting through the person ordained and set apart to use that power.
Of course non-Mormons will find the Mormon belief in priesthood power at least wrong, if not irrational. The non-Mormon religious will find it not only wrong but incredibly presumptuous: we say that God has delegated his authority to us and not to others; we say that certain works done by Mormon deacons, priests, elders, and others are different than the same works or similar works done by others because Mormons have priesthood given them by God. An LDS priesthood holder can do things—rites ("ordinances" in Mormon terms)—that others cannot. Through him comes a power to bind and loose on earth that non-Mormons do not have (Mt. 18:18).
I easily understand all three accusations by non-Mormons: wrong, irrational, presumptuous. I am sympathetic to the charges. Were I not a believer, I would almost certainly agree. But I cannot deny my experience. I trust what has happened through me and what has happened in my presence as well as the spiritual witness I have received. So I understand the charges well, but deny them. I have seen and felt the power of the priesthood.
Most other Latter-day Saints feel the same way, and because of that, not only is authority central to the LDS Church's self-understanding, it is central to the social fabric of Mormonism. Each local unit (a ward) is led by a bishop, called without ecclesiastical or theological training to serve without pay as the leader of the congregation for five years or so.
The bishop is called by the leadership at the wider level (the stake, roughly equivalent to a Catholic diocese). His calling is ratified by the members of his congregation, but they are not asked for input on that decision. They do not make suggestions to the stake president. They do not interview candidates to decide their suitableness for the congregation. They do not vote on a slate of candidates. (Something very similar is true of the stake president and for each leader up the ecclesiastical chain.)
Whatever the prior social or church position of the person called to be bishop, once he is called, members of his ward will show him considerable deference for the duration of his calling, and often beyond. We believe that he has been called by God to do the work of a bishop, and we honor that calling. Most members of the congregation will begin to call him "Bishop X" rather than by his first name or "Brother X," even if they have been on a first-name basis for a long time.
We don't often believe that the bishop was called by a direct revelation from God to the stake president. Most often, the stake president and his counselors make their decision (a decision that they submit to the Quorum of the Twelve Apostles for approval) a matter of prayer, thought, and discussion. They are likely to try to find out more about the person under consideration if they do not already know him well. They will consider the alternative candidates available. But in the end, they choose someone whom they feel they have been inspired to call.