Contrary to popular belief, Captain James Cook did not discover Australia.
The first non-Aboriginal people to visit (or 'discover') Australia were the Malay and Indian traders, from the Indonesian islands. They collected sea slugs from the Australian coast to trade with China, where the slugs were a prized delicacy.
The Portuguese are believed to have discovered Australia in the 1500s, but all records of their visit/s have been lost.
Willem Jansz/Janszoon was a Dutchman who was seeking new trade routes and trade associates. He became the first recorded European to step foot on Australia's shores on the western shore of Cape York Peninsula, on 26 February 1606. However, he believed the Cape to be part of New Guinea, from whence he crossed the Arafura Sea, so Australia was not charted as a separate continent at that stage.
In 1616, Dutch sea-captain Dirk Hartog sailed too far whilst trying out Henderik Brouwer's recently discovered route from the Cape of Good Hope to Batavia, via the Roaring Forties. Reaching the western coast of Australia, he landed at Cape Inscription on 25 October 1616. His is the first known record of a European visiting Western Australia's shores.
The first Englishman to visit Australia was William Dampier, in 1688.
James Cook (not a captain at this stage) charted the eastern coast and claimed it in the name of the British in 1770, and for this reason, Cook is often wrongly credited with discovering Australia. Captain Cook was on a scientific expedition to observe the transit of Venus from Tahiti when he continued west, coming across New Zealand and then continuing on until he reached the Australian mainland and charted the Eastern coast. Cook was the first European to sight and chart the eastern coast of Australia, which he did between April and August 1770, naming the land New South Wales. He explored much of the eastern Australian coast on behalf of Britain, which was looking to found new colonies given the looming probable independence of the American colonies.