This starship was sent to Vulcan in 2258 to respond to a distress call from the planet. When it arrived, it was destroyed by the Romulan ship Narada in the Battle of Vulcan. When the USS Enterprise appeared at Vulcan, it skimmed over the wreckage of the Mayflower. The wreckage of the Mayflower posed a significant navigational hazard to the Enterprise as the latter ship maneuvered itself through the battlefield. (Star Trek)
The Mayflower was not mentioned in dialogue, and its name was only seen briefly on a piece of wreckage. A sketch of the Mayflower was part of the Star Trek special "Starships" and it was identified as resembling a Miranda-class ship in Star Trek - The Art of the Film.
According to the novelization, the Mayflower was sent by Starfleet to investigate a 'lightning storm' in the vicinity of Vulcan. This storm marked the approach of the Narada.