Basically, Greece is in a crisis because it spent too much money and took in too little in tax revenue. Greece has a very large public sector and many of those workers are really not doing anything of great value. In addition, Greece's tax collection system is very poor and many people evade paying taxes.
To fund this, Greece borrowed money on international markets. When the financial crisis hit in late 2008, its economy was hurt and it was also unable to borrow more money because investors were wary of lending and because Greece looked like a very bad risk. It was soon unable to service the debt that it already had. This is what led to the crisis that is playing out today.