One cannot separate the Church of Jesus Christ of Latter-day Saints from it's founder, Joseph Smith. And I cannot begin to discuss him comprehensively here. There are many great histories of him. I'd recommend Richard Bushman's Joseph Smith: Rough Stone Rolling.
Joseph Smith tells his early story in one of the sections of The Pearl of Great Price. It is one of several different accounts of his early experiences. He tells that as a young man he lived in New York State, where there was "excitement" about religion, and how even his own family was divided about what church to attend. On a clear spring day in 1820, he, recalling the verse from James 1:5, retires to a grove of trees to "seek wisdom" directly from God. His intent was not to found a church, but simply to inquire which of the churches of his time he should join.
In the grove, he attempts to pray aloud, encounters the oppressive influence of a "power" of "darkness", and is, after struggling and fearing for his life, freed from the dark power and sees a pillar of light directly over his head. In the light are two personages. One calls him by name, points to the other and says, "This is My Beloved Son. Hear Him!"
He is then instructed to join none of the churches, but to wait for further instructions. Those do not come for three more years, when Joseph receives a visit from another heavenly messenger, Moroni. Moroni was the last prophet of the Book of Mormon and appears to Joseph three times during the night and once more in the morning, telling him of an ancient record buried in a nearby hill. Over the course of the next four years, Joseph is taught and prepared by this messenger in annual visits to the hill before he receives the record and finally translates it by the power of God.
It is not until 10 years after Joseph's first vision the grove that the church is officially organized, after the initial printing of the Book of Mormon, and after considerable persecution for Joseph and his followers.
Over the next 14 years, the church Joseph founded in a farmhouse in New York state grows to populate the city of Nauvoo, Illinois – the the largest city in the state. The band of followers has traveled to Kirtland, Ohio, various cities in Missouri, and finally to Nauvoo, driven each time by angry mobs who found reasons to oppose the growing band of saints.
In June of 1844, Joseph and his brother Hyrum were martyred while held in jail in Carthage, Illinois.
During his life, Joseph received revelations on doctrinal issues as diverse as the nature of God, the organization of the priesthood, the establishment of ordinances, the building of temples, a health code for Latter-day Saints, as well as personal instruction for many members. He translated the Book of Mormon and recorded his revelations in the Doctrine and Covenants. He was a civic and military leader as well as a religious one.
He was the first prophet of our dispensation. Through him the Lord restored His church, His priesthood authority, and the ordinances with which we can prepare ourselves to return to His presence. Mormons do not worship Joseph Smith, but they do honor him. My own experience is that as I have learned of Joseph and the fruits of his ministry, I have come to recognize him as the prophet that he was.