The eurozone debt crisis has been going on for so long now that I can't remember the last time I read a short precis of it which actually explained what the hell was going on to someone who hasn't been reading The Wall Street Journal every day for the last few years. So here is one.
The eurozone is weird. It's so weird that the closest historical analogy to it are the American states in the period immediately after the American Revolution, a political grouping that proved so unviable that it only lasted eight years before the ratification of the constitution and the creation of the United States of America. What the countries in the eurozone have in common with these early confederated states is that they have a common currency and individual economies which, if managed poorly, can cause disaster for the entire group, but no effective way for the group as a whole to ensure this doesn't happen.
This problem might conceivably have manifested itself in many different ways - and it was very different for the early American states - but in the eurozone today the problem has come to a head due to government debt. Each eurozone country has to borrow money individually rather than collectively to fund public spending, and as it needs to borrow euros, naturally it does so from within the eurozone itself. Greece, for instance, might borrow from German banks, Italian pension funds, and Spanish insurance companies. Through this process, the eurozone economy became much more integrated and interdependent, a process which it was hoped would bring the continent closer together politically as well as part of the general idea of promoting European unity. But it also increased the possibility of problems in one country eventually becoming a problem for everybody through the process we now usually call "contagion".
Say, for instance, that Greece were to run up debts that were so high that it was probably going to end up refusing to pay them back. What then would happen to all of those German banks, Italian pension funds, and Spanish insurance companies who had lent Greece money? They'd all lose it, and some would probably go out of business, which would be bad for the German, Italian and Spanish economies. And if the damage was severe enough, widespread panic might ensue and another credit crunch occur.
To prevent this from happening, the governments of those countries would have to borrow money themselves and pump it into their own economies to keep their banks solvent ("recapitalizing the banks") and prevent a complete collapse of the banking system as occurred during the Great Depression - basically a rerun of 2008. Except this time there are other countries in the eurozone - Italy, Portugal, and Spain especially - which are already having trouble paying off their debts, and which might themselves be then pushed into defaulting and spreading further chaos if they tried to borrow more money. And so the cycle would continue.
That, then, is the basic problem: soaring government debt on the one hand, and a possibility that the entire European economy will spiral downwards if the debt isn't repaid. The next question that obviously arises is how we got into this predicament. This is a fairly familiar story, and I refer you to my credit crunch write-up for the extended version. European governments have run themselves on an economic model that relied on cheap credit since the eurozone was formed, constantly borrowing new money to repay old debts on the assumption that new money would always be available; when the credit crunch hit and the it became harder to borrow money, they began to run into problems repaying their debts. To make matters worse, the credit crunch caused an economic slowdown and so reduced the amount of money governments could collect in taxes, further inhibiting their ability to service their debts.
The credit crunch having created a large black hole filled with the world's largest "IOU" sign in the heart of Europe's finances, the question then becomes: where is the money to repay it going to come from? There are two answers to this question, one of which is what European governments are doing at the moment and the second of which is what they're probably going to end up doing eventually.
At the moment, the solvent, richer eurozone countries have clubbed together to create a gigantic pot of money to lend at low interest rates to governments and banks which are in trouble to tide them over until the crisis passes. In the meantime, the countries with the worst debt problems are supposed to slash spending and increase tax receipts - a big problem for countries like Greece, as alex has explained - so that in the future they can get by without having to borrow so much money. Unfortunately, all of this austerity tends only to depress economic growth further and exacerbate the problem in the short term - as well as causing high unemployment and resultant social unrest - which means it's unlikely that this approach will work for long enough to solve the problem without something snapping.
To prevent something snapping, a much more radical solution may turn out to be necessary. This would be for the eurozone to follow in the footsteps of the American states and to completely pool their economic governance. This would mean that all European states would jointly become responsible for the debts of all the others, but they would also get to set tax and spending collectively as well - or, more realistically, the rich countries would tell the poor countries what to do and they'd have to lump it. The rich countries wouldn't just be lending the money to the poorer ones anymore, they'd be giving them it - and in return they'd demand a much more forceful say in how it was spent.
This would finally eliminate doubts about the solvency of the poorer countries - if you look at the total public debt of eurozone countries compared to the total size of the eurozone economy, the ratio is one of the smallest in the world - but run into probably immovable political obstables, not least being the almost complete lack of democracy in existing European institutions. German taxpayers would wake up to find that they're suddenly responsible formally and in perpetuity for paying the debts of countries they consider profligate and bordering on immoral in their approach to spunking money everywhere; Greeks would wake up to find that their grim neighbour to the north is suddenly able formally and in perpetuity to decide when they retire and how big their pensions are and everything else; and the two problems would feed off each other, with political elites unable to ever finally resolve the situation, not least because nobody voted for it.
And anyone who thinks such a situation could continue for long before the final collapse of the eurozone itself is deluding themselves.