Short answers, pulled from the story.
Athena erupted fully grown and armored from the forehead of Zeus after Hephaestus cleaved his skull open with a double-headed axe. This birth occurred when Zeus swallowed Metis, the goddess of wisdom, who was pregnant with their daughter to prevent a prophecy of a son more powerful than him. Athena leaped forth shouting a mighty cry that made the sky and mother earth shudder, fully armed with spear and shield.
Athena won the contest for patronage of the city by offering the first domesticated olive tree to King Cecrops. This gift provided wood, oil, and food, symbolizing economic prosperity and peace, whereas Poseidon's saltwater spring was undrinkable. The olive tree became so sacred that its destruction was considered a capital offense, and the city built the Parthenon on the Acropolis to honor her.
Hephaestus attempted to rape Athena but was pushed away, and his semen fell on her thigh before she wiped it off with a tuft of wool. She tossed the wool into the dust, which impregnated Gaia, the earth, leading to the birth of Erichthonius, an important Athenian founding hero whom Athena raised as her own son. The daughters of Cecrops went mad after opening a chest containing the child and hurling themselves off the Acropolis to their deaths.
Athena transformed Medusa into a hideous monster with serpents for hair after Medusa was raped by Poseidon in Athena's temple. The goddess struck Arachne with her staff and drove her to suicide after Arachne claimed her weaving skill surpassed Athena's in a contest. Arachne was then transformed into the first spider, while Medusa's gaze turned any mortal to stone.
Athena sided with the Greeks and influenced the outcome of the conflict by aiding heroes such as Diomedes and Odysseus. She overpowered Ares by striking him with a boulder and appeared to Hector disguised as his brother Deiphobus to persuade him to hold his ground before leaving him to face his fate alone. Her influence also extended to driving Ajax the Great insane to massacre the Achaeans' cattle, leading to his suicide.
During the Renaissance, Athena became the personification of chastity and wisdom in paintings by artists like Sandro Botticelli and Andrea Mantegna. Her image has influenced symbols of Western freedom, including the Statue of Liberty and Britannia, and a full-scale replica of the Parthenon stands in Nashville, Tennessee. Today, a gilded replica of Phidias's Athena Parthenos towers over the city, and the Great Seal of California bears her image kneeling next to a brown grizzly bear.