Star Trek was a TV show that ran from 1966 to 1969, and has spawned ongoing TV series, movies, books, toys, graphic novels, games, etc, set in its universe. Even a Barbie. The show's 60th anniversary was a few weeks ago. The model was for the NCC-1701 Enterprise - not to be confused with NGC 1701, a galaxy numbered for the show's ship.

The show crafted several versions to use for filming, from a small 100mm model to a large 3.4m version. The large model could only be filmed from one side, due to lighting cables entering from the other. The film stock was reversed to simulate the other side of the ship.

The original 3.4m studio model is a "living cultural object", and was itself a character in the show, voiced by Majel Barrett. The artifact was donated to the Smithsonian National Air and Space Museum, who has an online exhibit detailing what their world class conservationists did to restore and preserve it. Here is the display of the restored late 60's studio model.

In the 1970s, a write-in campaign from Trek fans convinced NASA to name the first Space Shuttle test vehicle after the show's ship. It was not capable of space flight, but it was used for 5 atmospheric flight tests. Many NASA engineers and scientists developed an interest in real life space exploration from watching the TV show as children. This is most of the show's cast in front of NASA's Enterprise in 1976.

Almost every online version has been remastered, but this TV intro looks original (youtube). This is what the studio model looked like in the show.

Lastly, here is a modern symphony orchestra in Prague (youtube) doing the famous Star Trek theme.