First off, national debt (debt owed by a central government) is in no way a new problem. In fact, every developed country in the world has at some point amassed some form of debt (owing another party). This goes off the old economic principle that “if you’re poor, you have to work hard to get rich, but if you’re rich, you have to be stupid to get poor,” which causes developed countries (a country considered to have a high level of development as decided by the International Monetary Fund) such as Canada, the US, the United Kingdom, etc. and their currencies to borrow money very cheaply, much more cheaply then a developing country such as Afghanistan or even Brazil. In fact, because all of these developed countries can borrow money so cheaply, it just makes sense to run a deficit (an amount by which the government revenue falls short) sometimes. That’s because in the long run, the economies of the developed countries will grow faster than the cheap debt they’re acquiring to grow their economies.
So we’ve established that debt is not necessarily always bad for nations. In cases like the United States in 1945, they saw a time where there was a debt that was almost 100% of its GDP (market value of all the final goods and services produced within a country), their deficit was almost 20% of their GDP (high by any standard), but that humongous amount of debt was followed by the biggest expansion in their economy of all time, arguably the largest and fastest economic expansion in the history of the modern world. 
Debt is not bad, but debt that you cannot repay is very, very bad. The key is that when developed countries borrow money, the rest of the World assumes that they’re going to be able to pay it back. This was true until early 2010, when the world found out that Greece, a developed country associated with a strong currency (the Euro), could not pay back its debt. This realization has made debt for Greece much more expensive, but since the new debt is so expensive it makes it relatively impossible for Greece to ever find a way to repay its debt, creating a vicious circle.
This caused more speculation about other developed countries not being able to pay back their debt, which raised interest rates (rate of interest that is paid by a borrower for the use of money that they borrow) which has caused the vicious circle to expand and cause more countries to have more defaults and higher interest, and that constantly repeating. Which would be, and is currently, very, very bad for the entire world. This whole problem is made even worse in light of the fact that the world is just now pulling out of a recession, in which there is less money coming into governments through taxes but government spending has to be at more or less the exact same level. All of this in the context of Greece, the problem being that with the Greek people doing all in their power not to pay their taxes, but still wanting to receive the same social benefits from the government. With no money going in, and it being harder to get outside money it became really hard for them to pay any money back and provide any money to pay back their debt. What ended up happening was that the IMF (International Monetary Fund, organization of 187 countries to better deal with economic issues) and Eurozone (a group of 17 European Union member states that have decided to adopt the euro as their currency) countries decided that they were going to give Greece a bailout (loaning to a country or entity to prevent it from failing), as long as they enforced strict austerity (a government policy of deficit-cutting, lower spending and a reduction of benefits and public services provided) measures within their own country.
This Greek bailout was followed by Ireland getting a bailout, then Portugal, and then a continued effort to try to save all of these countries. Unfortunately, we haven’t seen any success with these attempts which has caused a complete downward spiral that has pulled in all of the Eurozone countries and the rest of the European Union down with it. If Greece continues to fail, then the Euro fails along with it, which could possibly end the European Union as a whole. This is another complicated problem, but that’s another blog post. I hope that these four paragraphs have made you understand the basics behind the European banking crisis. Thanks for reading!