Demeter was also the sister of Hades, the Greek god of the underworld. Hades fell in love with her daughter Persephone, even though she was in fact his niece it was not uncommon for family and relatives to mate (like Zeus and Hera for ex.). Having been denied her hand in marriage by her mother, Persephone was kidnapped by Hades and lived with him in the underworld as his queen. However during the time Persephone spent in the underworld, Demeter, as the goddess of the harvest, turned her back on the harvest due to her depression of her missing daughter. Zeus, the king of the gods, saw that the earth was falling into chaos due to the mortals having no food, he sent Hermes, the messenger of the gods to the underworld ordering Hades to release Persephone so that Demeter could harvest the earth. Having no authority over his brother Zeus, he decided to let Persephone go. However Hades was clever; before she left Hades asked Persephone to eat 12 seeds from a fruit that way she wouldn't be hungry during her travel back to her mother. Persephone ate 6 seeds and left for her mother. When she arrived her mother was thrilled to see her. Persephone told her mother of her time spent with Hades including the seeds. Her mother realized what Hades had done. The seeds represented the 12 months of a year. Since Persephone had only eaten 6 she would have to spend 6 months with her mother but also 6 months with Hades. During the 6 months with Hades, Demeter would not harvest the earth. But during the times she spent with her mother, the earth would harvest again. This is where we get the four seasons. Therefore Persephone is known as the goddess of springtime.
In the pre crisis universe - Demeter came to the Amazons of Paradise Island to ask them to join the Gods against the forces of the Anti-Monitor. She provided the Amazons with magical armour to help protect them against their common enemy.
Post crisis, and Demeter is one of goddesses that bestow her favours on the Amazons of Themyscira. When Princess Diana was given life, Demeter stood alongside the other goddesses and Hermes to give the new soul godly gifts. Demeter granted Diana ' power and strength, like the earth itself'. Years later Demeter blessed New Themyscira with fertile lands.
During the War of the Gods, the Olympian Gods were drawn into a war between different pantheons. Demeter came to clash with her Roman namesake, Ceres. Millenia before, the Olympians were tricked by Uxas of Apokolips to split into two beings. The origianl to stay in Greece and the copy to be worshipped in Italy. This weakened the Gods but they stayed like this for millenia until the War of the Gods brought them into conflict with themselves. After the war subsided, the original Olympians, Demeter included, left this plane to travel the universe. it was years later on their return that they would once again merge with their Roman counterparts to become a single deity.
She has control over crops and agriculture, and will often punish those she dislikes with famine. She is also the Goddess of the seasons, so she can change the climate and state of the earth. She is also the Goddess of Motherhood, and she can bestow a good or bad pregnancy upon mortals. When she is with Persephone, the earth is warm and ripe, but when she is away with Hades, the world is dark and bare. The story is just one of the instances where Demeter's powers can affect the earth itself.