There are various technical aspects to what fuel is right for what engine, but when it is boiled down to basics it comes down to whether or not your engine "knows" what fuel you are using.
In older cars, with an engine which is fed fuel through a carburettor, the motor has no idea what is being used as fuel. The petrol flows through a small hole in the carburettor, goes into the engine and is burnt with the exhaust gases going out the exhaust pipe. The small hole, or jet, has been fitted to the carburettor with ordinary unleaded fuel in mind.
If you use ordinary unleaded fuel, it will be mixed with air at roughly the ratio of one part of petrol (by weight) to 14 parts of air (by weight) which burns "perfectly" to produce power without wasting fuel (running rich) or causing damage to the engine (running lean). If you use a premium unleaded fuel in the same engine it will flow through the jet at the same rate but because it has more "bang" for the same amount of fuel, your engine will run rich, that is using more fuel than it really needs.
On the other hand, if you have a later model car with fuel injection and an engine management computer, the motor effectively "knows" what is being burned as fuel. It does this by having an oxygen sensor, or lambda probe, in the exhaust system. If the mixture of air and petrol is perfect the probe will note it and send the information to the engine management computer. If the mixture is lean (lots of oxygen in the exhaust) or rich (little oxygen in the exhaust) the probe tells the computer which then adjust the fuel injectors to put more, or less, fuel into the engine to get the perfect burning mixture of air and petrol. So in effect the computer will make the engine use less fuel for the same performance and using a premium fuel in this system can be cost effective.
The vast majority of cars in Australia will run perfectly well on ordinary unleaded. High performance engines, especially those with high compression ratios, may need a premium fuel and other engines may produce slightly more power if a premium fuel is used.
Put simply, if your car has a carburettor, stick to ordinary unleaded. If your car is fuel injected with an engine management system (computer) using a premium fuel may be cost effective and may give more performance but most will be quite happy on ordinary unleaded.
Most of the fuel companies and car distributors have websites which carry more specific information.