In short:
Greece consistently spent more on social welfare (pensions, government subsidies, education, healthcare etc) than they could afford.
Normally countries would not be able to do this like Greece did, because running your economy that way has detrimental effects on your currency. But because they were using the Euro, the inflationary effects of their reckless spending were basically neutralised by the responsible policies of the rest of the EU. In effect Greece was reaping the rewards of the hard work and responsible economic policies of countries like Germany.
At present they owe over 320 Billion Euro to various creditors including several European countries and banks.
The only way they can repay it is if they start running their economy in a sustainable manner. That means reducing government spending and increasing tax revenue. Unfortunately the Greek citizens refuse to do this. They voted for a government that rejected these austerity measures, they voted in a referendum rejecting the notion of making the sacrifices.
The Greek people (obviously) don't want to pay more to live the way they do. While this is understandable it's also not realistic or feasible.
The situation for the Greeks atm is that their country has no money, which means that their people can't access money, which means they can't spend money. This is pretty disastrous. The ramifications for the owed nations is that they will probably have to write off the debt. This will hurt Greece far more than it's creditors.
Think of this analogy:
Imagine you have a relative who you have lent money to. You earn well, but your relative is struggling financially. They come to you and ask for a loan - to which you agree - because you can lend it to them much cheaper than the bank will (let's say you charge 1% interest but the bank would charge 10%).
Your relative uses this loan to live a lavish lifestyle - new cars, bigger house, parties, good food and wine etc. His wife spends the money on clothes, and jewellery, his family go on nice holidays, the kids have lots of nice toys. But when it comes time to pay you back, he asks instead for more money because he is struggling to make the car payments and the mortgage on his house. So you comply. But he continues to live lavishly, and instead of repaying his debts, he just keeps getting further into debt. His wife and kids are always finding new ways to spend the money and when he asked them to start cutting back they simply said "no".
Eventually this will catch up with him. You will put your foot down and say, no more. He will default on his car payments, his mortgage. So he will lose them. He and his wife and kids will have to suck it up and live a lot more modestly. And the longer they continue to live it up on money they don't have, the worse it will be when crunch time comes.