In Star Trek: First Contact, humanity’s first warp-capable starship is the Phoenix.
Zefram Cochrane builds the Phoenix from a refurbished nuclear missile in Bozeman, Montana. On April 5, 2063, he launches it and reaches warp speed, attracting the attention of a passing Vulcan ship. This event marks humanity’s official first contact with an extraterrestrial civilization.
The Phoenix is not the same vessel as any Federation starship named Enterprise. Its importance comes from its historic flight, which leads to Earth’s wider exploration of interstellar space. The film shows the ship making a short warp flight rather than beginning a long deep-space mission.
A common mix-up is confusing the Phoenix with Cochrane’s later reputation as the inventor of warp drive. The ship is the spacecraft that demonstrates his breakthrough; Cochrane is the person who develops the technology.