As I understand it, the LDS Church teaches that the Book of Mormon was written on gold plates in a language called "Reformed Egyptian". According to the LDS Church, what did Joseph Smith use and how exactly did he translate the Book of Mormon?
First, here's the simple version: Buried together with the golden plates was a device known as the Urim and Thummim, described by Joseph Smith as:
Several revelations, including the translation of the plates, were received by the use of these stones, through a process that is not well-understood.
Answering the question well, however is a bit complicated. By the time most people became aware of the Book of Mormon, the work of translation was in the past; both the translation and the publication of the Book of Mormon were completed before the organization of the Church or the missionary work that brought the book to so many people. And Joseph Smith himself never had much to say on the subject; he considered the book to be the significant thing, not the process by which it was produced.
The clearest and most detailed quote regarding the translation process comes to us from David Whitmer, an associate of Joseph Smith's, but one who was never actually involved in the translation process, so its authenticity is questionable. Whitmer stated:
Again, this explanation may or may not be accurate, as David Whitmer was never present to witness the process. Joseph's wife, Emma, who did serve briefly as a scribe during the translation process, gave a slightly different description:
Many years later, in an interview, she gave some more detail on the process:
The only things that are clear from the historical record, (by looking at surviving newspaper articles, personal journals, and the like,) are that Joseph Smith was widely believed by his contemporaries--both friends and enemies--to be in possession of a set of golden plates in the late 1820s, that he claimed that these plates contained an ancient record, and that he had the means to translate it by means of the interpreters (Urim and Thummim) found with the plates.
At one point, Joseph Smith produced a transcription of several characters from the plates, plus a translation of the sample, which was shown to scholars by Martin Harris, Smith's scribe at the time. He was skeptical of the work and wanted evidence. The record is uncertain as to the details of what happened when he did so, with Harris claiming that he received a favorable response, and Professor Charles Anthon, the scholar he showed it to, denying the claim. As Gordon B. Hinckley noted,
Unfortunately, while speculation abounds today, the actual concrete details of the process were never well-documented and have been lost to time, and all that remains are comments and recollections given years later.
Joseph Smith was surprisingly tight-lipped about the translation process. There isn't much we know about it.
Apparently, he had two seeing stones, Urim and Thummim. He used these for the translation of the first 116 pages from the golden plates. However, when someone stole the original translations, he claimed that the angel took away both these magical stones and the golden plates; he also claimed that he was not able to continue translation.
When the golden plates were returned to him, he used a different seeing stone that he had before he had found the golden plates. Amazingly, this magic seeing stone worked just like the other two seeing stones!
When, later in life, he was asked about all of this, all he said is that he was able to translate this mysterious language by 'gift and power of God'. At this point, either he was unwilling to repeat the story about the seeing stones or he was changing his story to not include the seeing stones. When his brother, Hyrum Smith, asked him for more details, Joseph replied that "it was not expedient for him to tell more than had already been told about the coming forth of the Book of Mormon, and it was not well that any greater details be provided".