Although Australia, Great Britain and New Zealand and other Commonwealth nations are democracies the public doesn’t have a direct say who will lead the country. To be Prime Minister you first need to be a citizen, then elected to the house of representatives, which is the lower house of the Parliament. The party you join has to then get a majority of elected representatives (seats). That party then elects the leader who will become Prime Minister.
The head of state of all three nations is the monarch of Great Britain who is currently Queen Elizabeth II. In Australia and New Zealand the Queen is represented by the Governor General who carries out her duties. The current Governor Generals are Quentin Bryce of Australia and Jerry Mateparae of New Zealand. The Governor General is not elected, rather the Prime Minister of the day makes the decision.