The Father and the Son appeared to the Prophet Joseph Smith

Joseph Smith's Testimony

"I saw a pillar of light exactly over my head, above the brightness of the sun, which descended gradually until it fell upon me .. When the light rested upon me I saw two Personages, whose brightness and glory defy all description, standing above me in the air. One of them spake unto me, calling me by name and said, pointing to the other�This is My Beloved Son. Hear Him!"
  ( Joseph Smith - History 1:17 )

When Joseph Smith told people about his experience it created some prejudice against him among those who professed to religion. Even though he was only an unimportant boy between 14 and 15 years of age, who was forced to make a scanty living from his daily labor, his person attracted attention among leaders of the different sects in the area at that time, to such a degree that it filled them with a strong spirit of opposition. Joseph testified:

"However, it was nevertheless a fact that I had beheld a vision. I have thought since, that I felt much like Paul, when he made his defense before King Agrippa, and related the account of the vision he had when he saw a light, and heard a voice; but still there were but few who believed him; some said he was dishonest, others said he was mad; and he was ridiculed and reviled. But all this did not destroy the reality of his vision. He had seen a vision, he knew he had, and all the persecution under heaven could not make it otherwise; and though they should persecute him unto death, yet he knew, and would know to his latest breath, that he had both seen a light and heard a voice speaking unto him, and all the world could not make him think or believe otherwise .. So it was with me. I had actually seen a light, and in the midst of that light I saw two Personages, and they did in reality speak to me." ( Joseph Smith - History 1:24-25 )

The Prophet Moroni hides the plates about 421 AD
A Heavenly Messenger appeared to Joseph Smith

Three years after God the Father and the Son appeared to Joseph Smith he asked about forgiveness for his sins and asked for more instructions from the Lord. As answer to his prayer a heavenly messenger, Moroni, appeared to him. Joseph told:
"He called me by name, and said unto me that he was a messenger sent from the presence of God to me, and that his name was Moroni; that God had a work for me to do; and that my name should be had for good and evil among all nations, kindreds, and tongues, or that it should be both good and evil spoken of among all people. He said there was a book deposited, written upon gold plates, giving an account of the former inhabitants of this continent, and the source from whence they sprang. He also said that the fullness of the everlasting Gospel was contained in it, as delivered by the Savior to the ancient inhabitants." ( Joseph Smith - History 1::33-34 )

Moroni was the last of the Prophets who wrote in the old record. About 421 AD the Lord commanded him to bury the plates with the writing of the prophets in the Hill Cumorah. Together with the plates there was a breastplate that contained the 'Urim and Thummim' - some seer stones which prophets of God used in ancient times to, among other things, receive revelations from God and to translate old holy records. Joseph would need these to translate the old record to modern language. The angel of the Lord showed Joseph, in a vision, where the old records were hidden. He also told him many important things pertaining to the Lords work in the latter days. The next day Joseph went to the Hill Cumorah. This is what he experienced in his own words:

"Convenient to the village of Manchester, Ontario county, New York, stands a hill of considerable size, and the most elevated of any in the neighborhood. On the west side of this hill, not far from the top, under a stone of considerable size, lay the plates, deposited in a stone box. This stone was thick and rounding in the middle on the upper side, and thinner towards the edges, so that the middle part of it was visible above the ground, but the edge all around was covered with earth. Having removed the earth, I obtained a lever, which I got fixed under the edge of the stone, and with a little exertion raised it up. I looked in, and there indeed did I behold the aplates, the Urim and Thummim, and the breastplate, as stated by the messenger." ( Joseph Smith - History 1:51-52 )

Four years of preparation would elapse before Moroni delivered the plates to Joseph. During this period Moroni provided him with further instructions about how the Church of Jesus Christ was to be organised on earth again.

The old records were translated to modern language by the gift and power of God
Joseph Smith translated these old records by the gift and power of God. Martin Harris and later Oliver Cowdery served as scribes in translation of The Book of Mormon. Oliver Cowdery testified on the events around the translation with the following words:

"These were days never to be forgotten�to sit under the sound of a voice dictated by the inspiration of heaven, awakened the utmost gratitude of this bosom! Day after day I continued, uninterrupted, to write from his mouth, as he translated with the Urim and Thummim, or, as the Nephites would have said, 'Interpreters,' the history or record called 'The Book of Mormon'." (History of Joseph Smith - footnote)

When the work with the translation were finished the first 5.000 copies of The Book of Mormon were printed. The Church of Jesus Christ of Latter-Day Saints were formally organised with 6 members in 1830. In 2018 there were 16.3 million members of the church worldwide. Source: Church News

Book of Mormon

"And now, after the many testimonies which have been given of him, this is the testimony, last of all, which we give of him: That he lives! For we saw him, even on the right hand of God; and we heard the voice bearing record that he is the Only Begotten of the Father." (D&C 76:22-23)