She was the daughter of the Titans Rhea and Chronus. So in Percy Jackson terms, she'd have been sister to the BIG THREE, Zeus, Poseidon, and Hades. She's symbolized by the 'horn of plenty' (Cornucopia? Hunger games anyone?) and was pretty much in charge of agriculture and fertility. All things that represent Spring and renewal.
Her daughter - Persephone, was kidnapped by Hades. In return, she forbade all plant life and growth while he held her prisoner. Once she was released, the green leaves and grasses resumed growth. So from this story, it was believed that Spring occurred when Persephone left the underworld, and Hades, and was returned to her mother for a short time.
Image and content from About.com Ancient History.