The federal government, based in Canberra and with the Prime Minister as head of government, makes laws for all of Australia e.g. Defence, Foreign Policy, Immigration, Marriage and Social Security. These responsibilities are broadly defined within the Australian Constitution.

State governments, with a Premier as head of government, make laws for the state e.g. Education, Health, Roads and Transport. The states are essentially responsible for those powers that are not granted to the federal government under the Australian Constitution.
Local government, headed by a mayor or shire president, is responsible for certain matters within the local council or shire area e.g. rubbish collection, recycling, local building permits and pavements. Local government is not included in the Australian Constitution.