The word "octane" has different meanings in different parts of the world. To understand how, you must first understand one of the physical properties of motor fuel.

One of the checmicals found in your fuel is isooctane, and what it does is helps the fuel to resist self-combusting when it's compressed. As you probably know, your engine works by compressing the fuel, igniting it, and then using the resulting "explosion" to push the piston back again and drive the crankshaft (in turn, in most engines that we're concerned with, this drives a piston to compress fuel in a different piston or pistons). Now gasoline's reasonably volatile, and if you compress it hard enough, it ignites all by itself: but isooctane helps to prevent that, allowing it to be compressed further without igniting.

There are a variety of different ways to measure how much isooctane is in a fuel, but they all boil down to basically the same thing: you put the fuel into a test engine that can vary the amount of compression it applies, and start burning it, gradually increasing the compression until the engine starts "knocking" (also known as "pinging"). This is what happens when fuel ignites prematurely, as a result of being compressed (i.e. before the spark plug fires), and it's not good for the engine nor for fuel efficiency. By measuring the pressure at which this happens, you can calculate the Research Octane Number (RON). Higher RON = more resistance to compression. If you drive anywhere in Europe or Australia, the number you're seeing (e.g. 95, 97) is the RON of the fuel.

Another way to measure the octane of a fuel is the Motor Octane Number (MON). This uses a different kind of test engine that proponents claim gives a better representation of the way that the fuel behaves under real-world conditions: at a higher temperature, faster engine speed, etc. The MON is always lower than the RON, usually by about 8-10 points (but exactly how much varies). In many countries with ratings for fuel, a fuel must meet a minimum rating of both RON and MON in order to be classified as "premium", but legislation varies.

When you drive in the US, though, what you're seeing is the Anti-Knock Index (AKI) (or Pump Octane Number). This is simply the mean average of the MON and the RON of the fuel. So broadly-speaking, a European 95 is about the same as a US 89, and a European 97 is pretty close to a US 93.

You should always use the fuel that your car manufacturer recommends. And here's why: if your car recommends regular fuel, but you put in premium fuel, it will have no appreciable effect whatsoever: the fuel gets compressed just the same, and is ignited by the spark plug just the same, and then provides the same amount of "push". In other words: premium fuel in a regular car is a waste of money.

The other way around is even worse: a premium-fuel car is tuned to compress the fuel more than a regular-fuel car. But when you put regular fuel in a premium car, there's a risk that the fuel will auto-ignite in the piston prematurely (i.e. before the right point in the stroke). This means that the "explosion" pushes backwards against the movement of the crankshaft - known as "pinging" for the noise that it makes. Using regular fuel in a premium-fuel car lowers fuel-efficiency and increases wear on the engine. Many modern premium-fuel cars with engine management computers can compensate for using the "wrong" fuel by adjusting the ignition time, but it'd still be better if you just put the right fuel in it to begin with.

That's a little bit of a simplification, but it's broadly what you need to know.