Mormons believe that we are led by a prophet, a claim that confuses some and outrages others. What does that claim mean?
When most people hear the word prophet, I imagine they think of someone who foretells the future. But that wasn't the primary function of biblical prophets, it isn't what distinguishes the prophets in the Book of Mormon, and it isn't what makes the prophet at the head of The Church of Jesus Christ of Latter-day Saints a prophet. It isn't that prophets never foretell the future, but that isn't what they do primarily.
Primarily prophets "speak forth," the etymological meaning of the Greek word from which we get the word prophet. They declare the word of God, which is sometimes a word about the future, but more often a call to repentance. A call to repentance may be harsh and direct (Hosea, Amos) or it may be gentle (Mt. 23:37). A call to repentance is something anyone can make. Anyone inspired by God to make that call is, therefore, a prophet.
But the LDS prophet not only speaks forth the things that God gives him to say, he does so in an authorized capacity: we believe that he is the one who is authorized by God to speak on God's behalf to the world and to the Church as a whole. Others can receive inspiration and revelation for their responsibilities, but he is the only one who can receive it for the Church and the world.
The LDS prophet's calling in that capacity doesn't mean that no other people receive divine inspiration. Indeed, leaders of the LDS Church have often said otherwise. For example, in 2002, reiterating a statement made 15 February 1978 by the First Presidency of the LDS Church (the highest governing council of the LDS Church), James E. Faust (at the time a counselor in the First Presidency) said:
[W]e claim that God's inspiration is not limited to the Latter-day Saints. The First Presidency has stated: "The great religious leaders of the world such as Mohammed, Confucius, and the Reformers, as well as philosophers including Socrates, Plato, and others, received a portion of God's light. Moral truths were given to them by God to enlighten whole nations and to bring a higher level of understanding to individuals. . . . We believe that God has given and will give to all peoples sufficient knowledge to help them on their way to eternal salvation." (First Presidency Message), Ensign, March 2002)
Mormons don't believe that we have a monopoly on divine inspiration.
Just as we make a mistake if we think that a prophet is someone whose main function is to foretell the future, it is a mistake to think that prophetic revelation is usually or always a matter of God whispering in the prophet's ear what the prophet is to say or revealing his will in a divinely given vision.
Such experiences are possible. Mormonism's first prophet, Joseph Smith, records both kinds of experience, as do later Mormon prophets. Nevertheless, most often prophetic revelation is much like the revelation others receive in their quest to know what God wills. It is Elijah's still small voice.
Continuing revelation through the prophet is a key part of LDS belief. It distinguishes Mormons from most other Christians. But the immediately previous LDS prophet, Gordon B. Hinckley said of it:
Now we don't need a lot of continuing revelation. We have a great, basic reservoir of revelation. But if a problem arises, as it does occasionally, a vexatious thing with which we have to deal, we go to the Lord in prayer. We discuss it as a First Presidency and as a Council of the Twelve Apostles. We pray about it and then come the whisperings of a still small voice. And we know the direction we should take and we proceed accordingly. (David Ransom interview, 9 November 1997)