Proserpina is an ancient goddess whose story is the basis of a myth of Springtime. She is the Roman equivalent of Persephone, which see for more details. She was subsumed by the cult of Libera, an ancient fertility goddess, wife of Liber. Her name comes from proserpere meaning "to emerge." She is a life-death-rebirth deity.
Venus, in order to bring love to Pluto, sent her son Amor to hit Pluto with one of his arrows. Proserpina was in Sicily, at the fountain of Aretusa near Enna, where she was playing with some nymphs and collecting flowers, when Pluto came out from the volcano Etna with four black horses. He abducted her in order to marry her and live with her in Hades, the Greek Hell, of which he was the ruler. Notably, Pluto was also her uncle, being Jupiter's (and Ceres') brother. She is therefore Queen of the Underworld.
Her mother Ceres, the goddess of grain or of the Earth, vainly went looking for her in any corner of the Earth, but wasn't able to find anything but a small belt that was floating upon a little lake (made with the tears of the nymphs). In her desperation Ceres' angrily stopped the growth of fruits and vegetables, bestowing a malediction on Sicily. Ceres refused to go back to Olympus and started walking on the Earth, making a desert at every step.
Worried, Jupiter sent Mercury to order Pluto (Jupiter's brother) to free Proserpina. Pluto obeyed, but before letting her go, he made her eat some pomegranate seeds (a symbol of fidelity in marriage) so she would have to live four months of each year with him, and stay the rest with her mother. So this is the reason for Springtime: when Proserpina comes back to her mother, Ceres decorates the Earth with welcoming flowers, but when in Fall she has to go back to Hades, nature loses any colour.
The myth of Proserpina, mainly described by the Roman Claudianus (4th century A.C.) is closely connected with that of Orpheus and Eurydice: it is Proserpina, as Queen of Hades, who allows Orpheus enter and bring back to life his wife Eurydice who is dead by snake poison. Proserpina played her cetra to quiet Cerberus, but Orpheus did not respect her order never to look back), and Eurydice was lost.
Proserpina's figure inspired many artistic compositions, eminently in sculpture (Bernini ), in painting (D.G.Rossetti , Pomarancio , J.Heintz , P.P.Rubens , A.Durer , Dell'Abbate , M.Parrish ) and in literature (Goethe  and Swinburne's Hymn to Proserpine)
For reasons that are quite obvious, a variety of pomegranate is called Proserpina.