How did Joseph Smith translate the Book of Mormon from the golden plates? The essay “Book of Mormon Translation” in the Gospel Topics section of LDS.org (topics.lds.org) gives insight into the process.

Translation Instruments

Joseph Smith and his scribes used two instruments in the translation process. When Joseph looked into the instruments, the words of scripture appeared in English.

1. One instrument, called in the Book of Mormon the “interpreters,” is better known to Latter-day Saints today as the “Urim and Thummim.” Joseph found the interpreters buried in the hill with the plates. Those who saw the interpreters described them as a clear pair of stones bound together with a metal rim, similar to a pair of “spectacles” or reading glasses.
2. The other instrument, which Joseph Smith discovered years before he retrieved the gold plates, was a small oval stone, or “seer stone.” As a young man during the 1820s, Joseph Smith, like others in his day, used a seer stone to look for lost objects and buried treasure. As Joseph grew to understand his prophetic calling, he learned that he could use this stone for the higher purpose of translating scripture.

These two instruments were apparently interchangeable and worked in much the same way, although Joseph often translated with the single seer stone rather than the two stones bound together.

Translation Mechanics

Some accounts by the scribes and others who observed the translation, say that Joseph studied the characters on the plates. However, most of the accounts speak of Joseph using either the interpreters or the seer stone by placing them in a hat and pressing his face into the hat to block out extraneous light. He would then read aloud the English words that appeared on the instrument.

Read the entire essay “Book of Mormon Translation” in the Gospel Topics section of LDS.org (topics.lds.org).

For more details on the translation process, see “By the Gift and Power of God” by Elder Neal A. Maxwell, Ensign, Jan. 1997, 36–41.