The Aborigines obviously discovered it first, thousands of years ago, as C says above.. For something a little more certain than the potentiality of the Dieppe maps and if you are referring to European 'discovery' in the 17th/18th centuries, there are various claims.
Dirk Hartog, a Dutch explorer, landed in Shark Bay in Western Australia in 1616. In 1642, Abel Tasman, another Dutchman, discovered the island of Tasmania - now part of Australia. William Dampier was the first Englishman to set foot there when he landed on the west coast in 1689. Captain James Cook's expedition did not get there until 1770, a century and a half after the first Europeans landed. For some reason, everyone seems to think he discovered the place! He was only the first to land on the east coast, to make any serious claim to the territory and to begin settlement of it.