Short answers, pulled from the story.
Mnemosyne was a Titaness, the goddess of memory, and the mother of the nine Muses by Zeus. Most sources name her as a daughter of Uranus and Gaia, though Hyginus's Fabulae instead calls her a daughter of Zeus and Clymene.
The name Mnemosyne comes from the Greek word mnēmē, meaning remembrance or memory.
Mnemosyne's nine daughters with Zeus were Calliope, Clio, Euterpe, Erato, Melpomene, Polyhymnia, Terpsichore, Thalia, and Urania. Each Muse governed a separate art, from epic poetry and history to dance, comedy, and astronomy.
Mnemosyne was worshipped in Lebadeia in Boeotia, at Mount Helicon in Boeotia, and within the cult of Asclepius. Statues of her also stood in the sanctuaries of other gods, including the shrine of Dionysos at Athens and the Temple of Athena Alea.
At the oracle of Trophonios in Lebadeia, described by Pausanias, supplicants drank from the water of Lethe to forget their thoughts, then from the water of Mnemosyne to remember what they saw during their descent. Afterward, priests seated them on the chair of Mnemosyne and asked what they had seen or learned.
Mnemosyne presided over a pool in Hades that served as a counterpart to the river Lethe, according to 4th-century BC Greek funerary inscriptions. Dead souls drank from Lethe to forget their past lives before reincarnation, while Orphic initiates drank instead from the river Mnemosyne to stop the soul's transmigration.
Mnemosyne is thought to have been classified as a Titan because memory was so central to the oral culture of Ancient Greece that it counted as an essential building block of civilization in the Greek creation myth. Titans were otherwise hardly worshipped and treated as archaic figures from the distant past.