From the definition of the word “republic,” it will be seen that without some qualification, the word is so imprecise as to be almost meaningless.
Opinion polls which ask peoples' opinion as to whether or not Australia should become “ a republic” are of little use. Australia was created as an indissoluble Federal Commonwealth under the Crown - a crowned republic. The question should define precisley which sort of republic is being proposed.
On this site we distinguish between the two major forms of republics.
One group are “crowned republics” ( also known as constitutional monarchies). At the centre of these is an institution above politics, the Crown.
The other category consists of “politicians’ republics”. In these there is no similar institution which provides leadership beyond politics.
In such politicians' republic the head of state ( and any state governor) is an elected politician, or one appointed by and controlled by the politicians.
These categories are not exhaustive.
Falling outside of these are, for example, absolute monarchies, which have existed historically in say, France under Louis XIV and exist today in Saudi Arabia.
Crowned republics, or constitutional monarchies, are ones where the monarch or Sovereign retains some of the formal executive and legislative powers of the state and provides leadership beyond politics.
The Sovereign, exercising the powers of the Crown, is an important check and balance to the political arms of the state. In a Commonwealth Realm, such as Australia, most of the powers of the Crown are normally execised by governors - general and governors who are appointed by The Queen on the advice of the prime minister or the premier. ( In Canada the provincial lieutenant governors are appointed by the governor-general. )
This role of the Crown provides a check and balance on the exercise of power by the elected politicians. Power is of course essential to government, but as Lord Acton famously warned, power tends to corrupt and absolute power corrupts absolutely.
In Australia, Canada, New Zealand and other constitutional monarchies, the Crown is important not for the powers which it wields, but rather, the power it denies others.
Under the Westminster system, the Crown constitutes the government advised by ministers who are responsible to the lower house of Parliament. This house is variously called the House of Commons, House of Representatives or Legislative Assembly.
The Westminster system has been singularly successful in assuring both stable and limited, democratic government.