Depending on your definition of “Europe”, there are 49 countries in Europe. Of these, only 18 use the euro.
These consist of the 13 “eurozone” countries (Austria, Belgium, Finland, France, Germany, Greece, Ireland, Italy, Luxembourg, The Netherlands, Portugal, Slovenia and Spain) plus Monaco, the Vatican, Andorra, San Marino and Montenegro. Kosovo, a disputed region of Serbia, also uses the euro.
On January 1, 2008, Cyprus and Malta will also adopt the euro. Slovakia, Lithuania, Estonia and Latvia all intend to adopt it over the next 2 or 3 years, and other EU countries (Bulgaria, Poland, Czech Republic etc) will also likely join further into the future.
Just to clarify what others have said, the euro is not an “either/or” currency. If a country uses the euro, that is its currency and (after a brief changeover period) its former currency ceases to be legal tender.
In non-euro countries, you cannot pay in euros, with the exception of a few tourist shops and perhaps border towns near the eurozone. If you tried to pay in a shop in, say, England or Sweden with euros youwould get short shrift, just as if you tried to pay in US dollars or Zambian kwacha.