The Repository in the hill Cumorah in New York (Mormon 6:6) was the site of all the Nephite records. Mormon gave his abridgment to Moroni, who added the abridgment (Book of Ether) along with his own comments when he "sealed" the record. Moroni put this set of plates into the stone box he constructed in a separate department of the hill Cumorah, as Orson Pratt explained long ago. This was the "original Book of Mormon" as Joseph Smith described it.
Joseph translated all of these plates (except the sealed portion) in Harmony, PA. He translated the title page there, which was on the last leaf of the plates. Then he and Oliver discussed going back to re-translate the Book of Lehi, the original translation of which Martin Harris had lost. Instead, the Lord told them they'd have to translate the plates of Nephi--except they didn't have the plates of Nephi. (D&C 10).
Joseph gave the Harmony plates to a divine messenger before leaving Harmony. After Joseph, Oliver, and David Whitmer arrived in Fayette, a messenger (probably the same one) brought the plates of Nephi from the repository in the hill Cumorah. That's why Joseph translated the plates of Nephi in Fayette.
This is important to understand because it explains otherwise inexplicable events in Church history. It also corroborates what Joseph and Oliver taught about the Hill Cumorah in New York, which renders irrelevant all the theories of Book of Mormon geography that put Cumorah somewhere else.