Premium gas has higher octane to prevent detonation, pinging, knocking. It does not increase horsepower. In fact, becasue it prevents detonation by burning colder than regular it actually contributes less Btu's and less horsepower. I only use it when it's called for by the manufacturer.
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.