(written from a Production point of view)
The Star Trek Smithsonian Exhibit was a Smithsonian Institution museum display opened at the National Air and Space Museum, Washington DC, running for a year from February 1992 through January 1993 in that institution. Actually, the exhibition – formally known as Star Trek: The Exhibit, or as simply Star Trek only, as indicated by the signage in the museum – was officially intended to be only a part of the larger, overall "Star Trek and the Sixties"-exhibit, as it was organized in conjunction with other displays, aimed at highlighting the interaction of Star Trek: The Original Series with other cultural phenomena of that era; yet these were utterly overshadowed by the Star Trek component. (Star Trek and History, Chapter 6) Discounting the "unofficial" science-fiction/Star Trek convention circuit, the specialized exhibit was as such the very first official – as in organized by professional parties – one of its kind, where Star Trek was concerned.
Shortly after the opening, the exhibit expanded in conjunction with the recent release of Star Trek VI: The Undiscovered Country, and production assets, most notably several studio models, from that production were added to the exhibit. Visitors to the exhibit had the chance to view the film on an omnimax film screen.
Having been the first large specialized Star Trek exhibit, garnering ample contemporary media coverage at the time, the exhibition turned out to be a runaway success. Smithsonian's former "Star Trek and the Sixties"-exhibit Advisory Curator H. Bruce Franklin has recalled in this respect, "When "Star Trek and the Sixties" opened, it turned out to be the most popular exhibition in the history of the Air and Space Museum, which had to issue tickets to control the huge influx of people [note: admittance to the Smithsonian museums is usually free, both in movement as well as in fees]. After more than a million people attended in Washington, the exhibition traveled to the Hayden Planetarium in New York City's American Museum of Natural History to be seen by another huge audience." (Star Trek and History, Chapter 6) The extended exhibition at the Hayden Planetarium opened in July 1993, running until 6 March 1994. At the planetarium, the exhibit was combined with the "Star Trek sky show", actually the Star Trek: Orion Rendezvous planetarium show as produced by the Oregon Museum of Science and Industry in 1992. 
While supported by it, it should be noted that this exhibit was neither produced nor organized by the official franchise, but a purely Smithsonian initiative. It was only after the resounding success of the exhibit that the franchise took over firm control over exhibitions, starting with the 1993 Star Trek Earth Tour.
It was not the first time the Smithsonian Institution recognized the cultural significance of Gene Roddenberry's creation; in a rare move – considering the highly contemporary nature of a television series of such recent date – , the Institution invited Roddenberry already in 1967 to submit the original pilots, "The Cage" and "Where No Man Has Gone Before" (the rare, original production variant, never seen again, aside from convention bootleg showings, after its submission to NBC at the tail-end of 1965, until the 2009 TOS Season 3 Blu-ray release), and assorted production material, such as still photography, scripts and story outlines, for save-keeping for posterity. This the consummate (self)promoter Roddenberry did in a formal presentation at the Institution, after the Original Series' first season had finished production, with the print materials showing up at the display twenty-five years later. ("Smithsonian Seeks TV Pilot", Los Angeles Times, 13 June 1967, p. C19)
While the exhibition was not particularly organized as such, a rough compartmentalization of its constituent parts could be made.
Set pieces, props and documentationEdit
As part of the display, set pieces and props from The Original Series were showcased, such as the original helm from the USS Enterprise and a tribble.  Original studio documentation involving the production of the Original Series, including the original script for "The City on the Edge of Forever", was also part of the exhibit. 
A special gallery was reserved to display for the first and only time, the Original Series garments as designed by William Ware Theiss, including Captain James T. Kirk's Starfleet uniform, with the gown worn by Leslie Parrish in "Who Mourns for Adonais?" being the eye-catcher.  Theiss' garments were sold off as his estate in the The William Ware Theiss Estate Auction pursuant the Smithsonian exhibit venue, and therefore halfway through pulled from the above-mentioned extension in New York City.
Especially noteworthy was the inclusion in the exhibit of the restored, actual studio model of the original starship Enterprise, the refit version from Star Trek: The Motion Picture (on loan from Paramount Pictures), the original model for the SS Botany Bay from "Space Seed" as well as the newly restored Class F shuttlecraft model from "The Galileo Seven", the latter two on loan from then owner Gregory Jein.  These models were not easily accessible to the public as they were suspended from the ceiling. Featured in display cases were the original, newly restored D7 class model, the K't'inga-class model (also on loan from Paramount Pictures) , and the original, restored Aurora model from "The Way to Eden". Originally one of the two original Tholian webspinner model from "The Tholian Web", the model was, like the original Enterprise and D7 models, gifted to the museum in 1973.  For the occasion, the museum had the restoration contractor for their possessions, Ed Miarecki, cast an additional copy of the model to represent the original Tholian webspinner as well, as was confirmed by Gary Kerr. (source) The actual second model was in effect still in existence, as it turned out to be retained by former Original Series set designer John Jefferies, who sold his possession at auction, nine years later. The exhibit marked only the second occasion (after the "unofficial" 1988 Los Angeles Equicon Science Fiction Convention) that a multitude of production-used Star Trek studio models, aside from the original Enterprise, were displayed to a general audience. For the museum's Tholian and D7 models, it has as of 2015 also remained the only time.
Aside from the already mentioned screening of The Undiscovered Country, the exhibit also ran several Star Trek documentaries made specifically for the exhibit. These documentaries included numerous interviews with Original Series cast members including more rare discussions with such guest cast members as Gary Lockwood and William Campbell. 
- "Smithonian Trek Photo-Feature", Peter Hardy, Sci-Fi & Fantasy Models, issue 25, January 1998, pp. 53-55
- "Star Trek Props: The 1992 Smithonian Exhibit", Star Trek: The Magazine Volume 2, Issue 10, February 2002, pp. 95-97
- "Star Trek At The Smithsonian", Star Trek: The Official Starships Collection, issue 129, July 2018, pp. 14-17