The octane rating of gasoline tells you how much the fuel can be compressed before it spontaneously ignites. When gas ignites by compression rather than because of the spark from the spark plug, it causes knocking in the engine. Knocking can damage an engine, so it is not something you want to have happening. Lower-octane gas (like "regular" 87-octane gasoline) can handle the least amount of compression before igniting.

The compression ratio of your engine determines the octane rating of the gas you must use in the car. One way to increase the horsepower of an engine of a given displacement is to increase its compression ratio. So a "high-performance engine" has a higher compression ratio and requires higher-octane fuel. The advantage of a high compression ratio is that it gives your engine a higher horsepower rating for a given engine weight -- that is what makes the engine "high performance." The disadvantage is that the gasoline for your engine costs more.
The use of a higher octane fuel than required, is a twofold mistake. You are wasting money and can possibly do damage to the engine. Using a higher than require octane rated fuel will not increase performance, clean the engine, or offer any benefits whatsoever. If your engine was designed for 87 octane, then use only 87 octane. If it requires 91 octane, then by all means use 91 octane. The engineers who designed the engine know full well what octane fuel it needs. The octane rating of gasoline determines how much the fuel can be compressed before it spontaneously ignites. That is all it does! (Look in your vehicle's owner's manual for information about the octane requirements of your vehicle).

Now, some people believe that using higher octane fuel will clean out your engine. However, higher-octane fuel contains no more detergent than low-octane fuel. If you feel you need some extra cleaning, there are additives that you can get at the local auto shop, but do not use them too much or it will damage your fuel injectors.