Demeter was the goddess of the earth, who was responsible for food and farming. She taught people the art of sowing and ploughing and so allowed the Greeks to live in cities rather than wander around collecting the fruits of the earth. She was a god of the country people.
Demeter was the mother of Persephone. When Persephone was taken by Hades, lord of the underworld, Demeter wandered the earth, and while she was doing this no grain grew. So Zeus sent Hermes to bring back Persephone to her mother. However, before she left, Hades gave her a pomegranate. When she ate from it, she was bound to spend a third of the year with Hades in hell. Now Demeter only lets things grow when her daughter is with her hence summer. The dying and blossoming of nature was thus connected with Demeter.
Demeter was shown as a solemn woman, often wearing a wreath of braided ears of corn. Here symbols are the fruits of the earth and the torch (from when she had to search for Persephone). Her sacred animals were the snake (an earth-creature) and the pig (another symbol of fertility).
As the goddess of grain and fertility, Demeter was worshipped with festivals and other honours.
The goddess Demeter was known as Ceres in Roman mythology.