Australia is the name of both a country and a continent. A country is a political construct. It is a territory that is ruled by a specific government. All the people in that territory are ruled by the laws of that government and are supposed to be protected by that government.
The boundaries of a country can change. For example, the United States changed markedly during the nineteenth century, growing from thirteen colonies on the East coast to a huge land mass that went from the Atlantic to the Pacific. In the twentieth century, the boundaries of Germany kept changing, depending on war and international political developments. As boundaries change, people in the same place can find themselves citizens of a different country.
A continent's boudaries, however, do not change. The word continent is a geographic description of a large mass of land that is distinct from other large land masses. Australia is a continent because it is a large land mass surrounded by the Pacific Ocean.
The country of Australia happens to comprise the entire continent of Australia. However, if the political situation changed, there is no reason why the continent couldn't be divided into several different countries. However, it would never stop being a continent, since that is its geographical designation.