The Greek goddess Hestia watched over domesticity and the family, and was honored with the first offering at any sacrifice made in the home. On a public level, Hestia's flame was never allowed to burn out. The local town hall served as a shrine for her -- and any time a new settlement was formed, settlers would take a flame from their old village to the new one.
As the equivalent of the Roman Vesta, Hestia was known as the virginal daughter of Cronus and Rhea, and sister of Zeus, Poseidon and Hades. She tended the fires of Mount Olympus, and because of her devotion to her duty as hearthkeeper, she managed to stay out of a lot of the shenanigans of the other Greek gods. She doesn't appear in too many of the Greek myths or adventure stories.
Hestia took her role as a virgin seriously as well, and in one legend, the lustful god Priapus tried to take advantage of her. As Priapus crept to her bed, planning on raping Hestia, a donkey brayed loudly, waking the goddess. Her screams woke the other Olympians, much to Priapus' great embarrassment. In some stories, it is said that Priapus believed Hestia to be a nymph, and that the other gods hid her by turning her into a lotus plant.
As a hearth goddess, Hestia was also known for her hospitality. If a stranger came calling and seeking sanctuary, it was considered an offense against Hestia to turn the person away. Those who followed her were obligated to provide shelter and food to anyone truly in need. It was also emphasized that female guests given sanctuary were not to be violated -- again, a grave offense against Hestia.
Hestia is traditionally represented by an image of a lamp with a perpetual flame. Today, some Greek reconstructionists, or Hellenic Pagans, continue to honor Hestia and all that she stands for.