The Off-Broadway Production of Hamilton was the first fully-staged production of the musical. It is, in essence, the earliest version of the production that eventually made it to Broadway, with the same creative team, characters, most of the songs, and the overall production design. The previous major iteration, the Vassar Workshop Production, was not fully staged and included songs that were cut from this version.
The production was staged at the Public Theater in the East Village (incidentally, on Lafayette Street) in the building's Newman Theater. It was extended twice before closing on May 3, a few months prior to the Broadway production's opening.
Contrary to some erroneous beliefs, no cast recording was ever made of this production: the leaked Soundboard Recording is actually a recording of a May 2014 lab of Hamilton (then still-known as The Hamilton Mixtape), and includes some songs that were already gone by the time the show opened at the Public. Due to the size of the theater, it is unlikely that a filmed bootleg was ever made of this production, although audio recordings have circulated over the years.