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There are generally believed to be 21 species of cockatoos:

Palm Cockatoo Probosciger aterrimus

Red-tailed Black Cockatoo Calyptorhynchus banksii
Glossy Black Cockatoo Calyptorhynchus lathami
Yellow-tailed Black Cockatoo Calyptorhynchus funereus
Short-billed Black Cockatoo Calyptorhynchus latirostris
Long-billed Black Cockatoo Calyptorhynchus baudinii

Gang-gang Cockatoo Callocephalon fimbriatum

Major Mitchell's Cockatoo Lophochroa leadbeateri

Galah Eolophus roseicapilla

Long-billed Corella Cacatua tenuirostris
Western Corella Cacatua pastinator
Little Corella Cacatua sanguinea
Tanimbar Corella Cacatua goffiniana
Broad-crested Corella Cacatua ducorpsii
Red-vented Cockatoo Cacatua haematuropygia
Yellow-crested Cockatoo Cacatua sulphurea
Sulphur-crested Cockatoo Cacatua galerita
Blue-eyed Cockatoo Cacatua ophthalmica
White Cockatoo Cacatua alba
Salmon-crested Cockatoo Cacatua moluccensis

Cockatiel Nymphicus hollandicus

Mitchell, T L
Major Mitchell’s Cockatoo Cacatua leadbeateri Vigors, 1831
[Alt. Leadbeater’s/Pink Cockatoo]

Lieutenant-Colonel Sir Thomas Livingstone Mitchell (1792-1855) was a Scottish army surveyor and explorer. He was the Surveyor-General of New South Wales from 1828 until 1855, and led various expeditions into eastern Australia, between 1831 and 1836 and to tropical Australia, from 1845 to 1846. A very life-like coloured plate of his cockatoo appears in Mitchell’s Three Expeditions into the Interior of Eastern Australia, published in 1838. A town in Queensland is also named after him. He was a hotheaded man and was the last person in Australia to challenge an opponent, in this case a politician, to a duel. Fortunately he only shot a hole in the man's hat, but honour was satisfied. He was heavily criticized for killing aboriginals, not only hostile tribesmen, but also harmless bystanders, such as old women and children. His actions cost the lives of members of his exploration party, such as his botanist Richard Cunningham. Mitchell said of the eponymous bird: few birds more enliven the monotonous hues of the Australian forest than this beautiful species whose pink-coloured wings and flowing crest might have embellished the air of a more voluptuous region.

Extract from Whose Bird - Common Bird Names and the People They Commemorate by Bo Beolens & Mike Watkins 2004 A&C Black ISBN: 030010359X