Hestia has no adventures. She is not abducted, does not quarrel, seduces nobody, punishes nobody, and appears in no surviving narrative of any length. For a figure in Greek myth this is close to unique, and it is not evidence of unimportance — it is the consequence of what she was. A hearth does not have a biography.
First and last of every sacrifice
The proverb Hestiai thyein, "to sacrifice to Hestia", meant to keep good things to yourself, because she got the first offering at every rite and nobody else saw it. The Homeric Hymn to her, one of the shortest in the collection, says that without her mortals hold no feasts, since it is her they pour the sweet wine to first and last. Every act of Greek worship was bracketed by her.
She is also swallowed first and disgorged last by Cronus, being the eldest of the six — which made her, in a formula the Greeks enjoyed, both the oldest and the youngest of that generation.
The fire in the prytaneion
The building that mattered was the prytaneion, the civic hearth-house at the centre of a Greek city, where a fire was kept permanently alight, where foreign ambassadors and citizens honoured by the state were fed at public expense, and where — in Athens — the Olympic victors and the descendants of the tyrannicides had the right to free meals for life. Socrates, in Plato's Apology, proposes dinner in the prytaneion as his appropriate "penalty", which was an outrageous joke precisely because everyone knew what the honour meant.
Colonists leaving to found a new city carried fire from the mother city's prytaneion to kindle the hearth of the new one. That is the single best illustration of her function: she is the continuity of a community as a physical, transportable flame, and a colony was not a settlement until the fire arrived.
The private version worked the same way. A newborn was carried around the household hearth in the Amphidromia ceremony, on the fifth or seventh day, which is when the child was named and formally admitted to the family. A bride was received at her husband's hearth. A suppliant who reached it could not be turned out.
The twelfth seat
The canonical twelve Olympians are a convention of art and civic religion rather than a fixed roster, and Hestia is where the convention shows its seams. She appears in some lists and not in others; Dionysus appears in the ones she does not. The Parthenon frieze includes her. A late tradition, first found in a Roman-era source, says she yielded her seat to Dionysus voluntarily to avoid conflict — a tidy explanation invented after the fact to reconcile two incompatible lists, and often repeated today as though it were an old myth.
Her vow of virginity has a similar look. The Homeric Hymn to Aphrodite names her as one of three goddesses over whom Aphrodite has no power, and says Poseidon and Apollo both courted her, and that she swore by Zeus' head to remain unmarried and was given a place in the middle of the house and the richest portion instead. That is essentially the only piece of narrative she has, and even it is really a statement about her cult privileges dressed as a story.
Vesta and the six women
Rome took her far more seriously as an institution. Vesta's round temple in the Forum housed no cult statue, only the sacred fire of the Roman people, tended by the six Vestal Virgins — the only female priesthood in Rome with full independent legal standing. They were selected as children, served thirty years, could make wills, gave evidence without oath, were preceded by a lictor, and could pardon a condemned man they met by chance. If the fire went out, the Vestal responsible was beaten; if a Vestal broke her vow of chastity, she was buried alive in an underground chamber with a little food, since her blood could not be shed and she could not be executed outright.
That is an extraordinary institution to attach to a goddess with no myths, and it lasted until Theodosius extinguished the fire in the 390s CE. The Roman round temple form itself is generally taken to preserve the shape of an early hut.
Why she has almost no modern presence
Modern retellings need plot, and she supplies none: she is absent from nearly every novel, film and game that uses the Greek pantheon, or appears briefly as a mild background figure tending a fire. The Percy Jackson books give her one of the few sympathetic modern treatments, as a quiet goddess at a hearth whom the hero notices when nobody else does.
Where she does survive is in the words. Focus is Latin for hearth, and the metaphorical sense arrives via optics. The Vestals gave English vestal for chaste. And the Olympic flame, lit at Olympia and relayed to each host city, is a twentieth-century invention — introduced for the 1928 and 1936 Games — that reproduces the colonists' fire-carrying almost exactly, without anybody having planned it that way.
A goddess with hearths in place of temples
Was she depicted in art?
Rarely, and never with a strong iconography. When she appears she is a modestly veiled, seated, fully draped woman, often holding a staff or a flowering branch, and she is usually identifiable only from an inscription or from the company she keeps. There was no widely copied cult statue type for her, unlike almost every other Olympian, and the Roman practice of housing the fire itself without an image points to the same idea: the goddess is present as the flame, not as a body.
Did she have temples of her own?
Some — Pausanias records a sanctuary at Hermione and a shared one with other deities elsewhere — but the prytaneion was effectively her temple in every city, which meant she had a building at the centre of political life in hundreds of communities and comparatively few dedicated shrines. Sparta's prytaneion, Olympia's, and the Athenian Tholos where the presiding councillors ate all served this role.
What is the Amphidromia?
A private family rite in which the newborn was carried at a run around the household hearth, usually by the nurse or the father, in the first week of life. It marked the point at which a child was accepted into the household — an acknowledgement with legal weight in a society that practised exposure of unwanted infants — and it was followed by a feast and by gifts from relatives. The circuit of the hearth is the act that makes the child a member of the house.
Is her name related to any English word?
Not directly. Hestia and Latin Vesta are usually taken as related, both going back to a root meaning "burn", though the details are disputed. The English hearth-words come from Germanic instead. What did travel is the prytaneion concept, through Latin and into the vocabulary of civic architecture, and the Roman priesthood's name in the adjective vestal.