NCC-1404 was the registry number of a Federation starship.
In 2368, it was one of the many decommisioned and adrift ships at Surplus Depot Z15. (TNG: "Unification I")
Background
The model used for NCC-1404 was one of the concept Excelsior models for Star Trek III: The Search for Spock. Although the model was labeled USS Alka-Selsior, it is most likely that this was a joke, and should not be taken seriously. The registry would seem to suggest an mid-23rd century starship.