A fuel's octane rating is the measure of it's resistance to detonation, that's all.

So a fuel with a higher octane enables you to tune the engine to a higher state, more compression or increased ignition advance (this may or may not make more power).
Or it gives you more leeway to be less than spot on with the jetting.

Avgas (well, the Avgas we get here) is very high octane (around 112) and has around 5 times the lead of old leaded 4* fuel and is here illegal to use on the road.
So it is very resistant to detonation, you can be numerous sizes too lean and it just loses power. The lead content is also a good thing for engine life.
However Avgas does burn quite slowly so although you end up running a lot more advance (we found we needed around 3° running 50/50 Avgas/97 RON UL) a lot of that advance is required as the burn is slower so the advance is not helping power, in fact the slow burn is making the situation worse as more of the compression cycle is being completed after ignition. This "negative work" is a bad thing.

Ultimately Avgas (or 50/50) has the potential to produce more power than normal pump UL however if you're running normal road bike head volumes (std NSR squish is a fairly slack 1.2mm) and stock ignition timing you will make more power on normal pump UL.
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