To reiterate and clarify the above points and more:
All EU countries and certain other countries comply with the first directive on motor insurance, which says that every insurance policy issued in the EU must provide the minimum insurance cover required by law in any other EU country.
Therefore all UK policies provide the minimum cover required by law in other EU countries (or the minimum cover required by UK law if that is greater). This cover doesn't automatically include theft or damage to your car. Most people want the same protection they have in the UK when travelling abroad, e.g. comprehensive or third party, fire and theft. Some insurers may automatically provide this extended cover for a specified period and often without additional charge, however it is important that you check with your insurer.
The Green Card is a document that is recognised in over 40 countries as proof that the minimum legal requirements for third party liability insurance in any country for which the Green Card is valid are covered by the insured's own motor policy. A Green Card is not required by law to cross borders within the EU and some other countries, however the countries that do need a Green card are:
Albania, Belarus, Bosnia and Herzegovina, Former Yugoslav Republic of Macedonia (FYROM), Islamic Republic of Iran, Israel, Moldova, Morocco, Russia, Serbia and Montenegro, Tunisia, Turkey, Ukraine.