When the Euro came into circulation, the countries that adopted it agreed to a single currency but not a single fiscal policy.
As such, the countries were allowed to carry on taxing/spending as before. Countries which were strong before the Euro continued to do well and countries which were weak before the Euro continued to do poorly.
This left a huge real-world discrepancy between the 'value' of the Euro in various parts of the union. Then the world economic crisis really screwed things over, with various European banks/countries struggling (Greece, Ireland, Portugal etc).
In order to save the whole single currency, those stronger economies (German and France) have had to lend more and more money to the weaker countries to keep them afloat.
However, it's become clearer and clearer that countries like Greece have basically taken the money and not fixed the issues with their economy which got them into trouble in the first place.
We're now at a stage where they are talking about writing off half of Greece's debt (meaning the banks and countries which own that debt being out of pocket) and we're looking at China getting involved.
That's in very simple terms and I'm happy for anyone to correct/amend anything I've said or missed.