How the Book of Mormon was translated
LDS Missionaries teach that Joseph Smith translated the contents of the golden plates by the "power of their god" and show this picture from the "Preach my Gospel" manual they are instructed to use.
Investigators are told that in order to know that the Book of Mormon is true, they must read, ponder, and pray about it. Moroni 10:3-5
Urim and Thummin
The Urim and Thummim is an ancient instrument or tool that Mormons claim has been prepared by God and was used by Joseph Smith to aid in the translation of the Book of Mormon (see Gospel Topics lds.org).
Emma Smith, the wife of the Mormon prophet, said that Joseph first used the Urim and Thummim, but later used a stone and hat. (Mormon Enigma: Emma Hale Smith, page 26).
Professor Anthon tells of Martin Harris' version of the translation through spectacles: "This young man (referring to Smith) was placed behind a curtain, in the garret of a farm house, and, being thus concealed from view, put on the spectacles occasionally, or rather, looked through one of the glasses, decyphered the characters in the book, and, having committed some of them to paper, handed copies from behind the curtain, to those who stood on the outside. Not a word, however, was said about the plates having been decyphered "by the gift of God." Every thing, in this way, was effected by the large pair of spectacles."
Translation by means of a hat
One of the Book of Mormon witnesses, David Whitmer, tells that Smith would put his face into a hat. In the darkness of the hat Joseph saw one character at a time with the English translation underneath it. (David Whitmer, An Address to All Believers in Christ, Richmond, Mo.: n.p., 1887, p. 12.) Russel Nelson, a Mormon Apostle, quoted Whitmer in an address given to mission presidents at the Provo, UT Missionary Center on 25 June 1992.
Emma, who also served as Josephs scribe during the translation process, confirmed this way of translation. She said that her husband would put a smooth dark stone in his hat and than lower his face into it to block out all light. (See Mormon Enigma: Emma Hale Smith, page 26.)
Verification of the Golden Plates Hieroglyphics controversial
Martin Harris claimed that Professor Charles Anthon confirmed the correctness of the translation and stated that they were Egyptian, Chaldaic, Assyriac, and Arabic. Harris also said that Professor Anthon gave him a certificate verifying the correctness of the charaters and translation, which the Professor tore up upon learning that the golden plates were given to Joseph by an angel. Read the whole story in LDS Church History 1.
Professor Anthon's story dramatically differs from Martins. He states that he warned Harris about, what to him, looked like a trick or hoax and declined giving him his opinion in writing. The characters presented to Professor Anthon by Martin Harris were: "Greek and Hebrew letters, crosses and flourishes, Roman letters inverted or placed sideways, were arranged in perpendicular columns, and the whole ended in a rude delineation of a circle divided into various compartments, decked with various strange marks, and evidently copied after the Mexican Calendergiven by Humboldt, but copied in such a way as not to betray the source whence it was derived. ... the paper contained any thing else but "Egyptian Hieroglyphics."
Elder Nelson told Mission Presidents in 1992 that the details of the Book of Mormon translation are still not fully known.