Higher octane fuels have a higher resistance to something called pre-ignition. Basically making it harder for the fuel to combust until it is ignited by the spark plug.
The only cars that need the high octane fuels are those with an engine that have high compression. (In the past these were mostly sports cars, however many manufacturers are placing high compression engines in other cars too). Some cars with high compression engines include the Honda Odyssey and Nissan Maxima. These vehicles also have something called a knock sensor to retard spark and minimize the chances of pre-ignition by starting the igntion process at a point that is less than optimal to prevent pre-ignition. Technically this allows vehicle manufacturers to advertise a higher fuel economy rate and bring up a company's fuel economy numbers by testing with the higher octane fuels.
Your owner's manual will inform you exactly which type of fuel you'll need for your car. Using fuel of a higher octane, typically will not hurt your vehicles engine, while using a fuel of a lower octane will cause what is known as "Engine Knock". An engine that has high compression, will actually have places inside the cylinder that is hot enough to begin the ignition process that is at a point other than the spark plug. These points of pre-ignition are typically where the piston and cylinder walls meet. This will cause a flame front to start at a point other than the spark plug. If the flame front when the spark plug starts will eventually collide with the flame front generated by the pre-ignition. When these flame fronts collide, you'll get the shock waves. (and you'll hear these shock waves in the forms of "Knocking")
If your owners manual says "Premium Fuel Required", it means no knock sensor has been added, and you must use the higher octane fuels to prevent engine knock. If you're owners manual says "Premium/Plus Fuel Recommended", it typically means you've got that knock sensor that will basically tell the electronics to retard spark and minimize/prevent knock by igniting the fuel at a point that is less than optimal, but preventing engine knock/damage.
If it says 87 octane, then using any of the higher octane fuels won't hurt the car, but typically you won't benefit in any way either.