Vesta was the Roman goddess of the hearth, the equivalent of the Greek Hestia. The names are linguistically related, and for the basic origins see Hestia. In a Roman household, much as in a Greek one, her shrine was the hearth, actual or symbolic, and Romans offered prayers to her before the main meal.
The attractively alliterative Vestal Virgins were Roman priestesses whose duty it was to attend the sacred hearth-fires. Traditionally there were six of them, serving in the temple of Vesta at Rome. They were elected by lot from a short list of twenty girls and officiated for at least thirty years: ten of learning and initiation, ten of actual ministration, and ten in turn teaching the initiates.
They really were virgins, and if they lost their virginity during their term of office the penalty was death by being buried alive. There seems to have been another Vesta of some kind, perhaps a kind of earth goddess, who was said to have been the wife of Coelus and even the mother of Saturn.
Her name, according to some, was nothing to do with Hestia's, but was from the earth being clothed (vestita) with plants, or else from the stability of the earth, so that sua vistat, 'it stands by its strength'!