Australia matches the definition of an island, as it is surrounded by water on all sides. However, Australia is more frequently referred to as a continent rather than as an island; it is one of the largest land masses on Earth, so it fits the definition of a continent.
There are four main characteristics required for a landmass to qualify as a continent, and Australia meets all of them. First, Australia's tectonic plates are independent from other continents. Second, Australia has flora and fauna that are unique from those on other continents. Third, Australia has cultures that are unique from those in other parts of the world. Finally, enough of the population in Australia believes it to be a continent.