A nuclear vessel was any ship powered by a Nuclear fission reactor. Typically such vessels were operated by military forces, such as the US Navy. That force maintained nuclear-powered aircraft carriers, one of which was the USS Enterprise.
In 1986, time travelers Pavel Chekov and Nyota Uhura were in search of a nuclear vessel from which to obtain high-energy photons which they needed to recrystalize the dilithium aboard their stolen Klingon Bird-of-Prey, called the HMS Bounty. They were able to obtain some from the Enterprise, which they found docked at the Alameda Naval Base. Chekov, being Russian, had difficulty pronouncing "vessel" while in search of the vessel, referring to such ships as "nuclear wessels". (Star Trek IV: The Voyage Home)