Deiphobe, the Sibyl of Cumae was an oracle. She was granted long life by Apollo, as many years as grains of sand she held in her hand, but she had forgotten to ask to retain her youth. With her aging she withered away and she was suspended in a bottle in the temple of Hercules at Cumae (near Naples).
The epigraph to The Waste Land comes from The Satyricon written in Latin by Petronius Arbiter. One of the translations at this site was done by Alfred R. Allinson who wrote in the introduction to his book:
Although reading the section of The Satyricon where this quote came from is not useful to novice readers of The Waste Land (it was just an off-hand remark) you can, if you wish, read section 48 of The Satyricon at this site.
"The longest and most important section [of The Satyricon] is generally known as the Supper of Trimalchio, presenting us with a detailed and very amusing account of a fantastic banquet, such as the most luxurious and extravagant gourmands of the empire were wont to exhibit on their tables."
In Virgil's Aeneid the hero, Aeneas, meets the Sibyl and uses her as a guide in a trip to the underworld to meet the shade of his dead father. In Virgil's account he was previously told how the Sibyl wrote Apollo's oracles on leaves which the wind could scatter. The Sibyl gave no help in reconstituting the oracle. Thus, just as in The Waste Land, assembling the fragments to mean something has the work of the reader. Aeneas, not wanting to have to piece together Apollo's oracle, asked the Sibyl to tell it to him orally.
This site has several sections from Bulfinch's Mythology which contain synopsis of parts of The Aeneid concerning Aeneas and the Sybil. Here they are:
This site has more about the Sibyl of Cumae