- According to Ship Recognition Manual, Volume 4: Starships of the Original Series Era, it was a Voroth-class vessel named VSS Kah'nath.
On the morning of April 5th, 2063, it was passing through the outskirts of the Sol system, when it detected Dr. Zefram Cochrane's warp signature during a test flight of the Phoenix from the vicinity of Earth, a planet which the Vulcan High Command had been monitoring for decades. As per Vulcan first contact protocols, they decided to land on Earth, to meet the person who flew the first warp ship of their world. They landed near a settlement in Montana.
The senior of its three-person crew exited the vehicle first, and offered the Vulcan salute, and "Live long, and prosper." Dr. Cochrane shook the Vulcan's hand, and said thanks. The Vulcans and Dr. Cochrane then went into the local establishment for beverages, and a presentation of the local music. (Star Trek: First Contact)
In the Mirror Universe, first contact proceeded very differently, as Cochrane shot and killed the T'Plana-Hath's captain. The assembled humans similarly executed the rest of the Vulcan crew, stormed the ship and raided it for its technology. (ENT: "In a Mirror, Darkly")