In the July 1993 Ensign article, "A Treasured Testament", Elder Russell M. Nelson quotes David Whitmer (an early member of the LDS Church and signatory to the 1830 statement, The Testimony of Three Witnesses) for a succinct summary of Joseph Smith's method of translating the language etched on the Golden Plates into the Book of Mormon:

"Joseph Smith would put the seer stone into a hat, and put his face in the hat, drawing it closely around his face to exclude the light; and in the darkness the spiritual light would shine. A piece of something resembling parchment would appear, and on that appeared the writing. One character at a time would appear, and under it was the interpretation in English. Brother Joseph would read off the English to Oliver Cowdery, who was his principal scribe, and when it was written down and repeated to Brother Joseph to see if it was correct, then it would disappear, and another character with the interpretation would appear. Thus the Book of Mormon was translated by the gift and power of God, and not by any power of man." (David Whitmer, An Address to All Believers in Christ, 1887, Chapter 1, p. 12.)

The image above depicts Joseph Smith translating the Book of Lehi, with Martin Harris (another signatory to The Testimony of Three Witnesses) serving as his scribe. Harris reported that Joseph, while translating, "would sit in a different room, or up stairs." (See Grant Palmer, An Insider's View of Mormon Origins, 2002, p. 10. See also, E.B. Howe, Mormonism Unvailed, 1834, p. 14, reproduced here and here).

No eyewitness account of Joseph Smith's translation process includes him reading (or making any other direct use of) the Golden Plates. Martin Harris, Emma Smith, and others stated that while Joseph dictated the text of the Book of Mormon, the plates were concealed in a box or under a cloth, or even buried outside in the woods or in the mountains. (See An Insider's View of Mormon Origins, pp. 2-4.)

Despite the existence of Elder Nelson's 1993 article, Joseph Smith's method of translating the Golden Plates does not seem to be widely known amongst members of the LDS Church. One reason for this may be the inaccurate imagery utilized in official LDS Church publications and presentations to depict the process. Click here and here for two examples of this. Recently, however, the LDS Church has published an anonymously-authored essay addressing this issue, here, and appears to have even removed many of the misleading images from its website.