Entrance exams in Australia can be roughly split into two categories: public and private.
While government schools by default do not require 'entrance' exams during primary and secondary education, many government-funded institutes of tertiary education (Technical and Further Education colleges and universities) require them, the content of which varies state by state. Nearly all of them are built into the state's education system as the 'endgame' of secondary education, the same as A Levels in England or the International Baccalaureate elsewhere.
In Northern Territory, students must meet the requirements of their chosen tertiary course when they take their Northern Territory Certificate of Education finals. In South Australia, there is the very similar South Australian Certificate of Education, tertiary admissions again based on finals performance. In Tasmania, there is (you guessed it!) the Tasmanian Certificate of Education. Western Australian pupils must pass the dedicated Tertiary Entrance Examination to enter tertiary institutions. In New South Wales, students take the HSC exams (Higher School Certificate). As of 2009, a national ATAR (Australian Tertiary Admissions Rank) is awarded to pupils across all of the above states, based on their performance in the various final exams, enabling national comparisons to be drawn.
In Queensland, there is no formal tertiary entrance exam process per se: students are ranked by their OP (overall position, similar to the concept of GPA in the USA), which is derived from their performance in all school work in all subjects. They do not receive an ATAR.
There are also a variety of specialised entrance exams. International students, or students who haven't recently been awarded an ATAR (mature students), must take STAT (Special Tertiary Admissions) tests that assesses a their verbal and quantitative competency. Students for whom English is not a first language also have to take the Written English STAT exam. Students wishing to study medicine must pass the UMAT (Undergraduate Medical Admissions Test) or the PQA, the Personal Qualities Assessment.
Many private schools have admissions tests, usually of their own making. The content and difficulty of these can vary considerably: be sure to seek specialist advice if unsure. If in doubt, hire a tutor who knows the exam structure to help your child get to grips with the syllabus.