Basically the RON or octane rating gives you an indication of the amount the fuel can be compressed before it spontaneously ignites - the lower the RON, the less compression it can take. Igniting from compression causes engine knocking - which can be quite bad for the engine.
One of the ways of increasing engine power (while keeping engine volume the same) is to increase the compression ratio of the engine - doing so means that you have to use a higher RON fuel to avoid knocking. So generally more powerful engines of the same volume need a higher RON fuel.
There doesn't seem to be any evidence that using a higher RON than what your engine was designed for gives you any added benefits that outway the extra cost.
Adding onto what gooseberry has said, ron is onli impt to ur car if ur car's ECU is programed to o/p the bhp/torque using the particular fuel. Meaning, traditionally high compression engines ie honda/suzuki 11.0:1 requires high octane petrol to prevent pinging or early detonation. Hence the higher the compression, the higher the oactane u may need. NA racing cars even require octane booster as they run insane levels of compression. The worse i knoe of, was a b18c type R which blew even running on v-power. The compression was above 12.5+:1.
However, if u take a look at force induction engines, they may require high octane petrol despite their low compression. This is because some force induction engines are tuned to o/p power/torque @ a particular octane rating. This is why like megaweb mentioned, top gear's test showed the WRX to o/p the highest figures @ the highest octane petrol.
This is also why japanese cars have two different models. JDM & export. Japan has one of the highest octance petrol available to consumers. Whereas the export models factor into consideration of the mass market out their. This is also why some of their performance breeds especially the WRX & type R, there's a difference between the JDM & export models in performance figures. They are tuned for different octane petrol.
So conclusion is.. most of our cars do not need high octane petrol. For fuel economy, it may be due to easier combustion of low octance petrol (i'm not too sure abt this). As for performance, there is no difference unless ur ECU is tuned to perform at high octane petrol or u are driving a car with high compression engine. Any differences felt are usually in our mind unless its proven
Dun forget, there are countries still running on ron 87, 91, & 95 being their highest.