High octane is required for some cars: e.g., a turbocharged car will likely require higher octane to reduce risk of detonation with higher charge temperatures. High octane is also required to get the optimal power out of a tuned car: properly advanced timing will enhance power and economy (obviously, not at the same time).
Some cars are capable of utilizing the highest octane available, depending on the ECU programming (this is not as common).
Some cars can tolerate a lower octane for an unspecified time by retarding timing. This reduces economy and increases emissions but will allow you to limp home on lesser fuel.
That said, check the owners manual and use the octane recommended. If you use octane that's too low, you're risking damage to your engine. If you use octane that's too high, you're wasting your money.
Follow-up with my specific example: my owners manual says 91 octane is required for daily operation with lower octane tolerable for short periods (2004 Subaru WRX). The gas stations near me only stock 93 octane so, technically, those extra two points of octane are wasted.
EDIT SOME TIME LATER:
I have sinced picked up an Accessport and reflashed the ECU to Cobb's Stage 1 specifications. With that new program, the ECU requires at least 93 octane so my previous remarks about two wasted points are no longer relevant.
The Accessport also provides real-time and average fuel economy by monitoring the actual fuel injected and the miles travelled. The measurements support the points in the first paragraph: the increased octane is allowing the car to produce more power at peak as well as higher economy when driven like a grown man with children and who should know better....
By the way, the car is a delight after the reflash. I was skeptical before and now am greatly annoyed with myself for not taking the plunge years ago.