Thanks for your question!
Before federation, the 6 states—Queensland, New South Wales, Victoria, South Australia, Tasmania and Western Australia—were British colonies. Each had been given the power to make laws for that colony by the British Parliament. Federation marked Australia becoming a nation on 1 January 1901, where these former colonies—now states—transferred some of their law-making powers to the new Australian Parliament.
Areas not listed in Section 51 are the responsibility of state governments. These state area include hospitals, police and roads. Local councils are not mentioned in the Constitution, although each state has a local government law that provides the rules for the creation and operation of councils. Sometimes the federal Parliament and the state parliaments may make laws about the same thing, for example, roads and health. However, section 109 of the Australian Constitution states that if the federal Parliament and a state parliament pass conflicting laws on the same subject, then the federal law overrides the state law. Section 122 of the Constitution also allows the Parliament to override a territory law at any time.
Discover more about the how the three levels of government works in Australia on the PEO website.