The eurozone had mostly been about the politics of integrating Europe. But not anymore. The economic consequences of the euro have come back to bite and now threatens the whole European project.
Back in the good old days, there was some economic rationale between the shared currency. It helped to build a common market among the European countries by making it easier for businesses to operate in different countries (although different regulations in each country do hamper this). The euro is also more stable as a currency compared to the currencies of many of the smaller members of the euro which helps to lower borrowing costs. These benefits have been used as a carrot to get countries to reform their economies and improve their public finances (crucially, the tough rules for applicants to the euro do not apply to those countries who already use the euro).
However, on the flipside of these benefits is that in terms of the exchange rate of the euro and in terms of monetary policy (i.e. interest rates), it is necessary to adopt a policy of one size fits all. So instead of each country having a currency which changes depending on their own specific circumstances or interest rates which could be set as each country saw fit, one exchange rate and one interest rate is applied across countries with very difficult economic conditions.
A lower interest rate for new members is one key result of this. The reason for this is there are risks involved in lending in a currency which is different to your own and especially when the exchange rate of the currency can change significantly or when lending to a country who can manipulate its own currency (both are usually true for members new to the euro). Also, poorer countries typically have higher interest rates as there are less funds that are available for lending and because investments in developing countries are typically more risky. And, for a long time (but not anymore), investors did not differentiate between the government bonds from the different countries in the eurozone.
So when a country (which is typically poorer) joins, it has a new interest rate that is typically too low for its economy but is set for the average conditions across the eurozone (a group of rich countries with low inflation rates). The low interest rates fuel a surge in borrowing and these funds tend to be used to buy property among other things. A booming real estate market triggers a rush to put up more buildings which also boosts the tax revenue for the government. But this lending spree can easily get out of hand and has to end sometime and often painfully (as can be seen in the papers with regard to Greece, Spain, Portugal, and Ireland but not Italy (see previous post)).
The political process that bought about the euro (refer to the posting above) did not set up rigorous checks and balances to make sure that imbalances did not build up. Furthermore, monetary policy focused solely on inflation of consumer goods rather than asset prices (i.e. the real estate market). And countries were not willing to give up control of their finances to European bureaucrats. In the end, if no one is in charge and there is a lack of rules, things are always bound to get out of hand. Not that politicians have done a good job at fixing that either. This is why some government such as Greece and Italy now have governments which are not run by politicians. The solution may be more economist in government.