Australia, a Commonwealth country, does not have a president. The nation does have a prime minister. The prime minister of Australia is Malcolm Turnbull, the head of the Liberal Party of Australia.
Technically, the prime minister serves by way of what is known as a commission from the Governor-General of Australia. The Governor-General is the representative of the Queen of Australia, who is Elizabeth II of the United Kingdom, and acts in her stead. In fact, the prime minister oftentimes is the head of the party with a majority of members of the Australian House of Representatives. In some cases, the prime minister is selected by a coalition of House members when no single party maintains a majority in that house of Parliament.