If someone owes you a large amount of money and can't pay there are a number of options open to you:
a) go to their house and take their stuff (there are legal restrictions on this);
b) take them to court;
c) ask them nicely to pay you back;
d) agree with them that they can pay some back now and some back later;
e) agree with them that they can pay some back and let them off the rest;
f) write off the whole debt and put it down to experience.
These are some of the options open to the people who have been lenders to the Greek government.
Who are these people? Banks, insurance companies, pension funds - any institution that was looking for a place to keep their money and earn a return on it (government debt is regarded as a safer place than, for example, the stock market).
Why lend it to Greece? Because governments are traditionally (and ironically) safe people to lend to and Greece debt gave them a decent return compared to other governments.
Why did the Greek government need to borrow? All governments borrow money from time to time - to pay for railway projects, new hospitals, the Olympics, or to make up the difference when tax revenues aren't as high as what they are spending. Of course, they need to pay it back and the theory is that as the economy grows they will earn the tax revenue to pay back the debt. Simples.
However, the Greeks borrowed a bit too much and these banks and pension funds were a bit too keen to lend to them. Consequently we are now looking at a number of options for Greece and each one has its own consequences. If you write off the debt then that could potentially put banks and insurance companies out of business or at the very least cause some damage to them (and they could include YOUR bank and the company that insures YOUR car). If you force Greece to keep paying then they might end up being unable to pay and then not pay any of it off. Or if you write off some of it, what message does that send?
A small but significant example of the problem is that large pharmaceutical companies are so worried about not being paid that they have been cutting the credit available to the Greek insurance fund which provides prescription drugs - equivalent to them cutting back on the amount of drugs they are willing to sell to the NHS without the money up front. The consequences of this are that patients with cancer or heart problems are having trouble getting the drugs they need and people are lined-up outside pharmacies trying to get life-saving drugs for ill relatives.
This is just one example of the current consequences. However, if Greece ended up having to leave the Euro, this would throw up even more issues. They would have to have a new currency, this would probably devalue against the Euro but what would happen to someone who had borrowed 100,000 original Euros to buy a house - do they pay back the bank (which could be German) in original Euros (which will cost that person a fortune) or do they pay them back in new Greek Euros (which would lead to a loss for the bank)? What about contracts with foreign suppliers which involve multi-million Euro agreements? Are these original Euros or new Greek Euros now? And if you were Greek, how quickly would you try to get your money out of the bank to keep the value of it as the original Euro?
This is why it's all a bit of a nightmare. I have heard suggestions that it the debt gets written off then the banks should seize the island of Rhodes as compensation. Interesting but probably illegal under international law. However, at least it's an idea.