The word literally means 'young maidens'. Greek nymphs were nature spirits, and are not considered to be immortal by all authors. In the case of the tree nymphs, by cutting down a tree you kill the nymph who lives in it. One could only escape the revenge of an angered tree nymph by begging for her blessings before cutting down her tree.

Dryads (Hamadryads): Oak Tree Nymphs
These nymphs lived only as long as the tree they inhabited lived.
Eurydice -- Orpheus' wife
Melia -- Mother of Amycus by Poseidon
Sagaritis -- Killed by Cybele when Sagaritis caused Cybele's companion Attis to lose his virginity due to her charms.
Syrinx -- Pursued by Pan, she was turned into a reed on the banks of the river Ladon. Pan turned her into different lengths, and fit her together into what we now call the pan pipes.

Naiads: Fresh Water / Fountain Nymphs
These nymphs lived for a long time, but were mortal. They had healing powers, but they would bring madness to those who angered them.
Arethusa -- A companion of Artemis

Meliads: Ash Tree Nymphs
Meliads were formed from Oranos's blood as it fell onto the earth after his incident with Cronos.

Melids: Fruit Tree Nymphs

Oreads: Mountain / Cave Nymphs

Oceanids: Ocean Nymphs
Daughters of Oceanus and Tethys
Asia -- Mother of Atlas, Prometheus, Epimetheus, and Menoetius (sometimes Tethys is given as their mother)
Callirhoe -- Mother of Geryon and Echidna.
Doris -- Mother of the Nereids.
Electra -- A companion of Persephone, and mother of Iris and the Harpies
Eurynome -- Took care of Hephaestus when Hera threw him off Olympus. She was also the first queen of Olympus, but was expelled by Cronos and Rhea.
Metis -- Zeus' first wife, and the mother of Athena.
Perseis -- Wife of Hyperion, and mother of Circe, Aeetes, Pasiphae, and Perses.
Styx -- A river in the underworld. All godly oaths are sworn to her. She is also considered to be one of the children of Nyx and Erebus.

Nephaeae: Glen / Valley Nymphs

Nereids: Sea Nymphs
The 50 Daughters of Nereus and Doris
Amphitrite -- Wife of Poseidon and queen of the sea.
Galatea -- The "Milkey-White Maiden". She was loved by Polyphemus and Acis, but loved only Acis. When Polyphemus killed her love with a boulder, she turned Acis into a sparkling stream.
Glouce -- ???
Orithyia -- She was abducted and raped by Boreas, and gave birth to Calais, Zetes, Cleopatra, and Chione.
Scylla -- Turned into a monster
Thetis -- Raised by Hera, but went against the goddess by taking care of Hephaestus (with Eurynome) after Hera threw him off Olympus. Thetis' son was prophesized to be more powerful than his father, so none of the gods would marry her for fear of being overthrown. She was the mother of Achilles, and could change her shape at will.

Camenae: Roman Nymphs of the Springs

Limnades: Nymphs of Lakes / Marshes / Swamps

The daughters of Atlas and Aethra. They were the sisters of the Pleides and of Hyas.

Other Famous Nymphs:
Acantha -- loved by Apollo, who turned her into a flower.
Adrastes -- Cared for the infant Zeus
Adrastia -- Cared for the infant Zeus
Adtasta -- Cared for the infant Zeus
Aegle -- ??
Amalthea -- Cared for the infant Zeus - sometimes thought to be a goat
Amphitrite -- Wife of Poseidon
Arethusa -- turned into a fountain
Britho -- ??
Britomartis (Aphaia) -- ??
Calaeno -- ??
Callirrhoe -- ??
Callisto -- Turned into a bear
Calypso -- Turned Odysseus' men into swine - daughter of Atlas
Clytie -- A water nymph turned into a Heliotrope for constantly watching Apollo
Cromme -- ??
Cynosura -- brought up the infant Zeus
Dryope -- Turned into a poplar tree - mother of Amphissus by Apollo
Echo -- She loved Narcisus, but as she could only repeat the last word she heard adn he fell in love with his reflection, he never returned her love. She died lonely, and only her voice remains today.
Egeria -- A water nymph who advised king Numa about religion
Eidothea -- daughter of Proteus - sea nymph - helped Menelaus get home to Sparta from Troy
Helice -- ??
Hesperia -- Guarded the Hesperian Apples
Ida -- brought up the infant Zeus
Leucothea -- ??
Lirope -- ??
Lotis -- daughter of Poseidon. When chased by Priapus, she called to her father, and he changed her into the Lotus tree.
Maia -- Mother of Hermes
Melite -- mother of Hyllus by Herakles
Minthe -- Mint plant
Nomia -- ??
Nysa -- brought up the infant Dionysus
Oenone -- ??
Piplea -- Turned Daphnis to stone
Sagaritis -- ??
Salmacis -- A fountiain nymph who loved Hermaphroditus. He did not love her back, but she prayed to the gods that they might never be separated, and she got her wish: they were combined to form a single being with both male and female characteristics.
Thoosa -- mother of Polyphemus
Tyro -- wife of Crethus & mother of Aeson, Pheres, and Amathaon. Also the mother of Pelias and Neleus by Poseidon.