The strangest birth on Olympus
Hesiod's Theogony tells it: Zeus, warned that his first wife Metis (whose name means cunning wisdom) would bear a child mightier than its father, swallowed her whole. The result was history's most consequential headache, resolved when Hephaestus split Zeus's skull with an axe and Athena emerged fully grown, fully armored, and reportedly shouting. Wisdom born from the head of power, already armed: Greek myth rarely subtexts this politely.
What she rules (and what she doesn't)
Athena and Ares split war between them, and the split is the lesson: Ares is bloodlust, the berserk charge; Athena is strategy, the phalanx, the fortification, the exit plan. The Greeks made their war-wisdom goddess also the patron of weaving, olive cultivation, and shipbuilding, because to them craft and strategy were one intelligence. Her symbols: the owl (still shorthand for wisdom), the olive tree, and the aegis, her shield bearing Medusa's head.
The famous myths
The naming of Athens: she and Poseidon competed for the city's patronage; he struck the rock and gave a saltwater spring, she planted the first olive tree. The citizens chose the tree (food, oil, wood) over the spectacle, the city took her name, and the Parthenon is her receipt. Arachne: per Ovid, a mortal weaver boasted she outwove the goddess; the contest went badly, and Arachne became the first spider, which is why spiders are arachnids. Odysseus: throughout the Odyssey she champions the mortal whose weapon is also cunning; the goddess of stratagems adopting literature's greatest schemer is the most consistent casting in Homer.
Rome, and where to meet her properly
Rome absorbed her as Minerva, which is why a certain strict, brilliant Hogwarts professor carries the name (the full HP mythology tour is here). For the wider pantheon: Greek mythology for beginners free, Edith Hamilton for the deep Greek shelf (compared honestly with ours), and Athena takes her daily turn among 30+ pantheons in 365 Days of Mythology, one myth a page (★4.7 across 37 Amazon ratings).
Common questions
What is Athena the goddess of?
Wisdom, strategic warfare, and craft: strategy rather than bloodlust (that's Ares), plus weaving, olive cultivation, and the practical intelligence the Greeks saw uniting all three. Her symbols are the owl, the olive tree, and the aegis.
How was Athena born?
Fully grown and armored from Zeus's forehead, after he swallowed her pregnant mother Metis and Hephaestus relieved the resulting headache with an axe, per Hesiod's Theogony. Wisdom emerging armed from power's own head.
Why is Athens named after Athena?
She won the city's patronage contest against Poseidon by planting the first olive tree, judged more useful than his saltwater spring. The city took her name and built the Parthenon on the win.
Who is Athena's Roman equivalent?
Minerva: same portfolio of wisdom, strategy, and craft. The name survives everywhere from owls on old coins to Professor McGonagall.