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?
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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".
The Church of Jesus Christ of Latter-day Saints announced today the publication of more primary records and artifacts related to the translation of the Book of Mormon which, remarkably, sheds more valuable light on the translation process.
To quote the press release:
So along with the complete printer's manuscript, the Church also published photos of one of the seer stones Joseph Smith used during the translation of the Book of Mormon. Not to be confused with the Urim and Thummim, this seer stone was not necessarily paired with another:
Its caption reads:
The translation process is still not very well understood, but the magazine article referenced above is already available online and contains a lengthy description of what we know about it, including the instruments of translation. That article (and The Joseph Smith Papers) are imperative reads for anyone seeking a rich understanding of the history.
There's an offical LDS page to answer your question here. I think that's your most solid and comprehensive answer. You can't get much more "according to LDS church" than their own web page dedicated to the topic.
It's broken into 4 sections: "By the Gift and Power of God", Translation Instruments, Mechanics of Translation, and Conclusion.
I think it's worth the read (about 2,000 words). It provides the church's standpoint about the how the Urim and Thummim was used, as well as what the Urim and Thummim referred to. It addresses Joseph Smith's loss of the ability to translate. It has links to related topics. It's pretty comprehensive and even includes some time period info about seer stones and their cultural and physical presence in early 1800s America.
There's hardly anything the Church of Jesus Christ is more eager about than that people read and study the Book of Mormon. You should have an easy time learning all you can about it from official church sources.