Intruder, that's politics and history for you. There's actually at least four different notions of "Europe" right now, and whether a country is in or out of one of these four depends on whatever treaty they signed up for.
First, there's "geographical" Europe. Basically everything west of, and including, the Baltic states, Poland, Slovakia et ceterea down to Greece. To a certain extent, the western most parts of Russia are also considered to be part of Europe.
Then there's the European Union. This is a treaty ("of Rome", and heavily modified with the treaty of Maastricht) between most (but not all) European states, which started as a treaty to stabilize the price of coal and steel after WW2. Today the EU is able to make laws that are automatically applied in all EU member states. It's got a council of ministers, a parliament and such. Most former East-European states are now getting in, and there's even discussions with Turkey about whether they are allowed in or not.
Then there's the EMU, I think it's called. It's the subset of countries from the EU that have swapped their local currency for the Euro. Sometimes called the Eurozone as well. In order to become part of this pact, I think you have to be a member of the EU and have to apply some very strict budgeting rules to your government finances.
Then there's the Schengen agreement. It's a treaty between several EU states (plus, strangely enough, a few non-EU states as well) to abolish internal border checks, essentially taking the free trade concept to the limit. (As an example, I drove from NL to France a few weeks ago, without stopping at the NL/Belgium border, and again without stopping at the Belgium/French border. I also did not stop in Belgium itself for any reason - my wheels were turning the whole time.)
The UK is member of the EU, but is not part of the Eurozone and also not part of the Schengen area. But Iceland, on the other hand, is, as far as I know, not part of the EU, is also not part of the Eurozone, but did have a free trade agreement with various Scandinavian states before the Schengen agreement was signed. They did not want the trade restrictions back, so Iceland subscribed to the Schengen treaty. Denmark is all the way in, but hasn't swapped their local currency for the Euro. Instead they have a fixed exchange rate and on a more practical level, you can pay everywhere in Euro in Denmark anyway.
Wikipedia, as always, has the definitive list of names that are in each treaty. Being European, it still boggles my mind to remember who subscribed to what.
http://en.wikipedia.org/wiki/Europe
http://en.wikipedia.org/wiki/European_union
http://en.wikipedia.org/wiki/Eurozone
http://en.wikipedia.org/wiki/Schengen_Agreement
Oh, did I mention NATO already? Most European states, plus the US?
http://en.wikipedia.org/wiki/NATO