Essentially what happened was that nations of very different fiscal structures and situations (some with massive debts and others with manageable debts) came together under a monetary union with 1 currency, the Euro. Some of the countries with greater debts, like Italy had set up plans to reduce the debt over time and things worked fine when the market for their government bonds was good, but as world credit markets faltered, their rates rose to the point where their plans to reduce the debt no longer worked since they relied on lower borrowing rates vs. what they were paying before joining the Euro. Things have gotten so bad as everything spiraled that investors are asking for more and more return on the bonds which are getting riskier and riskier. This is making it even more expensive to pay off debt- to the point where some couldn't even afford it anymore (i.e. Greece hence the bailout). Now usually a country can inflate their currency to help pay off debts as inflation essentially acts as a tax and since the majority of bond obligations are in nominal terms, not real (so they can just pay back the debts with inflated currency). In this situation, however, there is only 1 currency and while countries like Greece and Italy would benefit from currency devaluation, other countries like Germany and France not only do not need it, but do not want that inflation. Since the latter two have much more power in monetary policy, the PIIGS, the countries that are really in trouble like Greece and Italy, are stuck in a bind where they can't pay off their debts.
I left out quite a few things and this is a simplified account, but it is a basic overview.