Short answers, pulled from the story.
Demeter was the ancient Greek Olympian goddess of the harvest, agriculture, grain, and the fertility of the earth. She also presided over sacred law, the cycle of life and death, and was a central figure in the Eleusinian Mysteries, which promised initiates a happy afterlife.
According to the Homeric Hymn to Demeter, Hades abducted Persephone from a meadow with Zeus's permission to make her his wife. Demeter searched for nine days before learning what happened. She then refused to let crops grow, causing a famine, until Zeus ordered Hades to return Persephone. Because Persephone had eaten pomegranate seeds in the Underworld, she had to spend part of each year there, explaining the cycle of seasons.
The Eleusinian Mysteries were a major religious ceremony held at Eleusis in Attica, centered on Demeter and Persephone. Unlike many ancient Greek rites, they were open to initiates of any gender or social class. The central theme was the reunion of Persephone with her mother, understood as a symbol of renewal and eternal life. They promised initiates better hopes in both this life and the afterlife, and their roots likely reached back to the Mycenaean period before 1200 BC.
Even after the emperor Theodosius I banned paganism throughout the Roman Empire, Greek communities continued to pray to Demeter as 'Saint Demetra,' patron saint of agriculture. Around 1765-1766, the antiquary Richard Chandler documented locals at Eleusis still covering a caryatid statue with flowers to protect their crops, calling it Saint Demetra. The statue was removed in 1801 by Edward Daniel Clarke and eventually donated to the University of Cambridge, where it now resides in the Fitzwilliam Museum.
The meaning of Demeter's name is debated. The second element, meter, derives from the Proto-Indo-European root for 'mother.' Scholars disagree on the first element: some connect it to the word for earth, making her 'Mother Earth'; others, like Jane Ellen Harrison, argue it relates to grain, making her 'Grain-Mother'; and M. L. West proposed the name was borrowed from an Illyrian source related to a Messapic goddess named Damatura.
Erysichthon was a king of Thessaly who ordered Demeter's sacred grove cut down to build a palace extension. When his men refused to touch a great oak hung with votive offerings, the king cut it himself, killing a dryad nymph. As punishment, Demeter sent Limos, the spirit of unrelenting hunger, into his stomach. The king ate everything he owned, sold his daughter Mestra into slavery, and eventually consumed himself.