Demeter and her daughter Persephone are Greek goddesses who are usually paired and found in connection with the Eleusinian mysteries. They are fertility goddesses, with Demeter especially important as goddess of grain and seasonal change. You see the association between Demeter and grain in the name of her Latin equivalent, Ceres, whence "cereal".
Another Greek name for Demeter is Kore
or maiden. Among the Romans she was known as Proserpina, which is sometimes rendered Proserpine, in English.
Triptolemus enters the picture from the story of the abduction of Persephone by Hades. Demeter was hunting for her missing daughter when she arrived at Eleusis and was welcomed into the king's household. After an abortive attempt on Triptolemus' brother -- when Demeter tried to do what Achilles' nymph-mother did to him: dose him with immortality -- Persephone taught agriculture to Triptolemus.