In 1606, Dutch explorers became the first Europeans to "discover" Australia, naming it New Holland. They did not settle it.
In 1688, English pirate William Dampier also came across Australia's western coast, but found the land to be inhospitable, and the Aborigines a "savage" people. He did not recommend Australia for settlement.
In 1770, British explorer James Cook (not yet a captain) encountered the east coast of Australia. He called it New South Wales, and claimed it for Britain. HE thought it would be a good place to settle.
In 1788 the British sent a fleet of ships to settle Australia. Britain needed a place to send its convicts (people who have been sent to jail for thefts and other crimes). This First Fleet settled at Port Jackson, establishing a settlement which they called Sydney. For the first few years they did not have much food, and life was very hard. But soon they began to farm, and more people came. Sydney grew, and new towns were started.
For twenty five years, the Blue Mountains to the west of Sydney prevented the colony from expanding. Once the explorers Lawson, Blaxland and Wentworth found a suitable crossing route over the mountains, there was no limit to the expansion of Australia. New colonies were created north, south and west.