Unlike most other Christians, Mormons believe that God is continually providing new revelations to humans, both personally on an individual level and generally for the broader Christian Church (which, for Mormons, is only the LDS Church). Mormons don't believe that the Bible is the only 'Word of God,' they believe that God has communicated equally important information that is contained in other texts.
Mormon Sources of God's Revelation
In addition to the Bible, Mormons believe that the Book of Mormon, Doctrine and Covenants, and Pearl of Great Price are all holy scripture. What's more, Mormonism teaches that the Bible has become corrupted, is filled with errors, and had important truths removed. Since those other (Mormon) texts have not been corrupted, the Bible is therefore inferior to them as a source of revelations from God about what God wants and information about God.
In addition to these texts, Mormons also believe that LDS President and Prophet receives divine revelations. His interpretations of those revelations are not only on an equal level with past communications from God, but can in fact supersede them. God's revelations to humanity are ongoing; just as the New Testament supersedes the Old Testament and Mormon scriptures supersede the Bible, recent revelations to the LDS Prophet supersede everything else.
This belief that the Bible has been superseded by more recent revelations from God of equal or higher authority is something that Mormons actively avoid discussing publicly with non-Mormons. Instead, they are more likely to be seen talking about the importance of the Bible in order to establish common ground with traditional Christians. You'll never see Mormon politicians like Mitt Romney saying that the Bible is defective or corrupt and that the Book of Mormon is more important.
Prophet and President
Mormons believe that Joseph Smith "restored" the original Christian Church, a church that was lost when the last apostles died. In other cases Christians who present themselves as restoring an original or true Christian church focus on "original" beliefs, dogmas, practices, etc. Joseph Smith, in contrast, focused more on an original "spirit" — including the spirit of ongoing communication with and revelation from God.
Thus the LDS President and Prophet is akin to the original apostles who followed Jesus and were responsible for developing the Christian Church after Jesus died. They were empowered with both spiritual and temporal authority, in large part because of their personal contact with Jesus.
The LDS President and Prophet claims the same sort of authority — the authority of the first Christian leaders like Peter and Paul. This authority includes determining how to interpret all scripture as well as the revelations he himself receives. There are no individual, personal interpretations of any of the texts.
The original Christians were the "saints" of the former days; Mormons today are the "saints" of the latter days. Their belief in ongoing divine revelation is, in many ways, not just the foundation of Mormonism but also the what most separates them from every other religious group in the world, including especially traditional Christianity.
Past vs. Current Revelations
Mormons are not the only ones who believe that God continues to communicate to people nor are they the only ones who believe that God might communicate new truths to people. Such beliefs are not uncommon at all among various Christian denominations. What really sets Mormons apart from traditional, orthodox Christianity is the belief that new, valid revelations can supersede previous revelations.
In effect, Mormons believe that God can change his mind, sometimes even radically, and thus overturn past practices and traditions in favor of something completely new. Most new revelations aren't quite that radical, of course, but some have been. Polygamy is a good example — both its introduction and its elimination. It's no coincidence that polygamy was one of the most serious sources of conflict between Mormons and Christians when Mormonism first got started.
Belief in ongoing divine revelation is not limited to just the LDS President and Prophet. He's the only one who can claim the authority to receive revelations that are applicable to the management and doctrines of the entire Mormon church, but all Mormons individually look for and expect divine inspiration or guidance for their daily affairs — family life, employment, business, etc.
In some ways this isn't remarkable because it's common for Christians to pray for guidance or for a sign, but in Mormonism this approach to one's relationship with God seems to be taken much more seriously and pursued much more actively than in other religious movements.
Unlike others, Mormons already accept that God is actively guiding their entire church with new information. Mormons also believe that God was once a human just like them, making the distance between them and God much smaller than it is in traditional, orthodox Christianity — especially since they believe that Mormons can becomes gods as well.
Mormon Texts and Leaders on Divine Revelations
In Mormonism's Articles of Faith, the ninth Article of Faith says:
"We believe all that God has revealed, all that He does now reveal, and we believe that He will yet reveal many great and important things pertaining to the Kingdom of God."
In Doctrine and Covenants (43: 2-3) it states:
"For behold, verily, verily, I say unto you, that ye have received a commandment for a law unto my church, through him whom I have appointed unto you to receive commandments and revelations from my hand. And this ye shall know assuredly—that there is none other appointed unto you to receive commandments and revelations until he be taken, if he abide in me."
Ezra Taft Benson wrote in 14 Fundamentals in Following the Prophets:
"Beware of those who would pit the dead prophets against the living prophets, for the living prophets always take precedence."
Benson also wrote:
"The Bible sits on the pulpit of hundreds of different religious sects. The Book of Mormon, the record of Joseph, verifies and clarifies the Bible. It removes stumbling blocks, it restores many plain and precious things."
Bruce McConkie writes:
"There is no more false or absurd doctrine than the sectarian claim that the Bible contains all of the word of God."