Gas is expensive right now, and where I live, the difference between regular and premium is about $0.20/gal. So to save money, I put regular gas in my car (Volvo, naturally aspirated). The owner's manual says to run premium. You can do the same thing, to save money.*
The main difference between "premium" and "regular" gasoline is the octane rating. The octane rating number is a measure of how resistant to premature detonation (knock) the fuel is. Some premium fuels may have additional detergents and other additives. For a more detailed article on gasoline chemistry, see Wikipedia.
Knock occurs when the air/fuel mixture in the cylinder detonates due to heat and pressure, before the spark ignites the mixture, or ahead of the flame front after a spark ignition. The result is faster than desired combustion, and higher than normal cylinder pressure, which can damage the engine. Knock sounds from the drivers seat like a metallic rattling or pinging.
High performance engines typically run higher compression ratios, or forced induction, since more air and fuel per L means more HP/L. The higher the compression ratio, the higher peak pressure and temperature the air/fuel mixture will reach during the compression stroke, and the more knock resistant the mixture must be.
Modern cars have an accelerometer (or two) mounted on the engine block, to detect knock. This accelerometer measures high frequency (5-20kHz) vibrations of the engine block. On a V8 engine, you may have one knock sensor on each engine bank. As your car runs, the engine controller (ECU) constantly monitors the signal from the knock sensor, and uses it to adjust the timing of the ignition spark. Under normal conditions, the ECU will advance the spark timing, moving it later in the compression cycle, until it senses the very early stages of knock ("borderline spark"), minus a safety margin. This maximizes the power output.
If the ECU senses knock, it attempts to prevent it by retarding the timing of spark, moving the start of ignition earlier in the compression cycle. This reduces the peak pressure in the cylinder, reducing the likelihood of knock. It also reduces the power output of the engine, because less pressure means less torque.
This is what will happen if you run regular gas instead of premium. You will pay for this with slightly worse fuel economy, as you might loose 5% of your power, but you may not even notice the difference. Labeled fuel economy is calculated from controlled drive cycles in a lab, and few of us drive consistently enough to measure small differences in fuel economy.
* To be safe, I would not run regular gas in a turbocharged or supercharged engine. It does not take much knock at high load and high RPMs to ruin your day. If you do hear knock, slow down your engine right away (tip out), drive gently, and go back to premium gas.