Best results are running the lowest octane you can without getting pre-detonation.

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 name "octane" comes from the following fact: When you take crude oil and "crack" it in a refinery, you end up getting hydrocarbon chains of different lengths. These different chain lengths can then be separated from each other and blended to form different fuels. For example, you may have heard of methane, propane and butane. All three of them are hydrocarbons. Methane has just a single carbon atom. Propane has three carbon atoms chained together. Butane has four carbon atoms chained together. Pentane has five, hexane has six, heptane has seven and octane has eight carbons chained together.

It turns out that heptane handles compression very poorly. Compress it just a little and it ignites spontaneously. Octane handles compression very well -- you can compress it a lot and nothing happens. Eighty-seven-octane gasoline is gasoline that contains 87-percent octane and 13-percent heptane (or some other combination of fuels that has the same performance of the 87/13 combination of octane/heptane). It spontaneously ignites at a given compression level, and can only be used in engines that do not exceed that compression ratio.

During WWI, it was discovered that you can add a chemical called tetraethyl lead to gasoline and significantly improve its octane rating. Cheaper grades of gasoline could be made usable by adding this chemical. This led to the widespread use of "ethyl" or "leaded" gasoline. Unfortunately, the side effects of adding lead to gasoline are:

Lead clogs a catalytic converter and renders it inoperable within minutes.
The Earth became covered in a thin layer of lead, and lead is toxic to many living things (including humans).
When lead was banned, gasoline got more expensive because refineries could not boost the octane ratings of cheaper grades any more. Airplanes are still allowed to use leaded gasoline, and octane ratings of 115 are commonly used in super-high-performance piston airplane engines (jet engines burn kerosene, by the way).

As far as I've been told, you can increase/decrease your timing (to a point) to take advantage of higher/lower (respectively) octane fuels. If an engine at a constant compression and timing gets a shot of higher octane fuel, it will need a hotter spark to ignite the mixture since it doesn't have enough compression to raise the flash point to the same level as the lower octane fuel. If the ignition can't provide the needed spark, a misfire occurs and economy and power suffers. Conversely, a higher octane fuel at a given compression will need less spark heat to initiate combustion...in some case no spark at all...which would be knock. Changing the timing alters the effective compression at the point of ignition.