Will higher octane fuel give me more power?

[AaRoN]

The answer....No, it won't.  In fact, the only real benefit that you will get from it is alittle more cleaning.  Higher octane fuels (91,93) have more detergents in them. 
Why? You ask....
Well, higher octane fuels are more resistant to combustion.  So, if your vehicle says "Regular Unleaded", it was designed (compression ratios, spark plug type and vehicle operating temp all come into play here) to be most efficient with 87 octane.  By efficient, I don't just mean MPG's either.  Efficient as in; emissions, MPG and power.  If you use say, 93, in a car designed for 87 you'll have improper combustion with alot of leftover fuel.  This means build-up in your combustion chamber.  Deposits are bad and rob performance.  You won't notice a difference for awhile. MPG will go into the toilet first and then performance will begin to drop.  Basically what's happening is, your combustion chamber (cylinder) isn't getting hot enough to completely ignite all the fuel being sent in.  Another bad side effect is that you'll begin to clog your cats and foul your O2's. 
Using 87 in a car designed for 93 can and will most likely cause pre-ignition.  The 87 octane "burns" easier than the higher octane fuels so it ignites too soon.  If your car has a knock sensor, the PCM will get a signal from it saying to retard the timing.  Loss in power amongst other problems.