Mormon is a colloquial term that originated in the 1830s as a reference to people who believe that The Book of Mormon is divine scripture, who are members of The Church of Jesus Christ of Latter-day Saints, or are members of one of the many smaller organizations that separated themselves from that Church.
According to Church theology, Mormon was a prophet who lived in the Americas circa 400 A.D. Faced with the destruction of his people, the Nephites, by the Lamanites, Mormon compiled a history of his people on gold plates. This history, which was an abridgement of more extensive records, was passed on to Mormon's son, Moroni, who buried them in order to prevent them from being destroyed by the Lamanites. Latter-day Saints believe that the location of the plates was later revealed to Joseph Smith by the angel Moroni and translated into English by divine assistance. This translation is published as The Book of Mormon.
The use of the word "Mormon" to describe Church members, as well as the Church itself, arose soon after the publication of the Book of Mormon in 1830. Although originally used pejoratively, the term has been used widely within the Church, as evidenced by the name of the Church's most prestigious choir "the Mormon Tabernacle Choir," the names of several Church songs, such as "I'm a Mormon Boy," and the tagline, "From the Church of Jesus Christ of Latter-day Saints, The Mormons," which appeared at the end of a series of Church-produced television commercials. However, as early as 1979 Church leaders began discouraging use of the term due to concerns at least in part that the term "Mormon Church" was being used to the exclusion of the Church's official name, which is believed to have been revealed by God (see for example  (http://library.lds.org/Library/lpext.dll/ArchMagazines/Ensign/1979.htm/ensign%20may%201979%20.htm/we%20the%20church%20of%20jesus%20christ%20of%20latterday%20saints.htm?f=templates&fn=document-frame.htm&q=Church%20Name&x=Simple&2.0#LPHit1)). In 2001, the Church issued an official statement regarding the name of the Church and the use of the term "Mormon" in the media (see  (http://www.lds.org/media2/library/display/0,6021,198-1-168-15,FF)). It encourages use of the official name of the Church or, as an abbreviated form, "the Church" or "the Church of Jesus Christ", and advises that the term "Mormon" or "Mormon Fundamentalist" is not properly applied to other religious groups, especially those now practicing polygamy. Notwithstanding, the Church has since acquired and set up a website at http://mormon.org intended for non-Mormons looking for more information.
Despite the Church's efforts to encourage use of its official name, the Associated Press has continued to recommend "Mormon Church" as a proper second reference in its influential Style Guide for journalists. Additionally, some scholars feel the term "Mormon" is useful to collectively describe all those groups which derive from Joseph Smith. Sometimes "Restoration Churches[?]" is used for this purpose instead, but that can lead to confusion since there is an entirely different group of Christian churches (those derived from the Campbellites or Stone-Campbell churches, for example, the Church of Christ[?] and the Disciples of Christ) which are also called Restoration Churches.
Some scholars (e.g. Melton, in his Encyclopedia of American Religion) subdivide the Mormons into "Utah Mormons" and "Missouri Mormons". The Missouri Mormons are those Mormons who did not travel westward to Utah, and the organizations formed from them (the RLDS, Strangites, Temple Lot, etc.), while the Utah Mormons are those who did travel westward to Utah, and the organizations formed from them (The Church of Jesus Christ of Latter-day Saints and the various polygamy-practising groups who are separated from the Church). This nomenclature is not common within the Church and the use of "Mormons" or "Mormon Fundamentalists" to refer to organizations or groups outside of the Church is a grave misunderstanding of Mormon theology, in particular the principle of continuous revelation and Priesthood authority.
Official information about The Church of Jesus Christ of Latter-day Saints (Mormons): http://mormon.org