A democratic government is one in which the citizens of that government have a say in who their government is composed of. This traditionally takes the form of elections where candidates for the different positions in government campaign, and using that information citizens vote on which candidate best represents their beliefs and values.
Australia is part of the commonwealth, a collection of sovereign states that are part of the remaining British Empire. Consequently, the government of Australia is a constitutional monarchy with Queen Elizabeth II as head of state. What this means is that the head of state, a component of the executive branch of government, is actually the monarch of the United Kingdom, presently Queen Elizabeth II.
Australia is a constitutional monarchy – 'constitutional' because the powers and procedures of the Australian Government are defined by a written constitution, and 'monarchy' because Australia's head of state is Queen Elizabeth II. (Australia.gov)
The Australian government is a parliament that is divided into two chambers, the Senate and the House of Representatives. Australia is a democracy such that the citizens of Australia vote for their Senators and members of the House of Representatives during elections.
In this way, Australian citizens indirectly choose their Prime Minister. The party who holds the majority power in the House of Representatives selects the party leader who will be named Prime Minister.
By convention, the Prime Minister is a member of the House of Representatives ... (Australian Parliamentary Elections Office)