The European Union began as a common market for goods and services. A common European currency space is a more recent development -- the Euro serves as the currency of most (but not all) EU Member States. Adoption of the Euro has reduced trading costs, and has led to more transparent prices.
The Euro crisis is first and foremost a sovereign debt crisis, initially affecting a handful of EU Member States running unsustainable deficits. The sovereign debt crisis is itself an artifact of the establishment of the Euro -- neither Greece nor Spain would have been able to borrow as much in their former currencies (or on such favorable terms) as they were able to using the Euro. Prior to the Euro crisis, the financial markets valued Euro-denominated Member State obligations similarly. As the crisis developed, lenders became far more discriminating, demanding much higher Euro interest rates from weaker Member States (such as Greece and Spain) than from others.