Australia is a constitutional monarchy and our head of state is the Queen. However, the Queen does not have a role in the day-to-day running of Australia. On the advice of the Prime Minister, the Queen appoints the Governor-General, who is the Queen’s representative in Australia.
The Australian Constitution delegates—gives—certain powers to the Governor-General to act on behalf of the Queen. These include giving Royal Assent to laws passed by the Australian Parliament and starting the process for a federal election. While these powers are exercised by the Governor-General, in reality this is normally done on the advice of the Prime Minister and ministers.
For more information on the role of the Queen, have a look at the overview of the Australian Constitution by the Australian Government Solicitor and this fact sheet about Australia's system of government.