Vesta was the Goddess of the Hearth, Family, and Home in Roman belief. Hestia was her Greek counterpart.
Vesta was the sister of Jupiter, Juno, Neptune, Ceres, and Pluto. At one time an Olympian, but she gave up her throne to (most say) Bacchus, who was the God of Wine and was Vesta's nephew (being Jupiter's son). Vesta didn't mind this at all, as she was really happier tending her hearth than sitting on a throne.
Vesta was something of the deities' black sheep, for she wasn't argumentative, loud, trouble-making, or otherwise "difficult". Indeed, she was a quiet, calm woman; some say she didn't even have a distinct personality! She didn't have any inclination to marry or any desire to give up her virginity, so she had no lovers or children. She simply spent most of her existence tending the divine hearth as well as the hearths of mortals.
Because of her low-key nature and monontanous job, there aren't many tales about Vesta's adventures (or lack thereof, really). However, she was widely worshipped by the Romans, for she watched over life-sustaining hearths as well as the families and homes that contained them.
The Vestal Virgins were Vesta's special priestesses. Their most important duty was to keep Vesta's holy fire burning; if a Virgin let the sacred fire go out, she would be executed.