Lets start with explaining what exactly is European Union(E.U.) and European Central Bank(E.C.B). To understand that we need to know what are monetary and fiscal policies.
Monetary Policies are the policies which are related with actions of central bank of a country i.e it determines the rates at which banks lend money, banks borrow money from central bank and intrest rates banks pay to depositors.
Fiscal Policies are what we call budgets of governments. Fiscal policies determine how much each country spends on different industries and areas of development.
(Generally a deficit budget is considered good for a country looking to develop.)
Now lets get back to EU. After the devastation of World War II, the fastest way to develop Europe was to get down trade barriers. This was done by building a centralised unit (E.C.B) which controlled Monetary policies of whole Euro Area(common currency i.e, Euro was introduced in 1999).
IT IS TO BE NOTED THAT FISCAL POLICIES WERE UPTO INDIVIDUAL GOVERNMENTS (which allowed them to borrow according to own wishes)
So EU had members which were economically very strong e.g, Germany. Inclusion of strong economies gave weaker economies greater access to credit which they cannot repay unless their economy grows and expands rapidly. However, this is not what happened with Greece, their economy has shrunk 25% in last 5 years and it has loans, about 180% of their GDP.
The money which should have been used for development of Greece has been used by political parties for own benefit and paying out pensions.
Now how did EU allow Greece to borrow so much?
After the 2008 collapse of Wall Street(Lehman brothers and others) credit was tightened and all the loans Greece was taking to pay back previous loans were suddenly stopped . In 2009 Greece announced that it had been understating its figures .
In 2010, Greece gave in to inevitable and was going towards bankruptcy. E.C.B bailed them out during that period with conditions of cutting spending and following strict borrowing measures. However this did not go as planned as lower spending meant less tax collections as incomes were dropping .
This has brought us to the time when Syriza is elected as Government. They were brought to power on promises of getting Greece out of debt and improving its position.
Greece government has publically voiced opinion of not getting a good deal out of negotiatons with creditors ( saying that acceptance of terms would mean disaster for Greece) and wants to opt out of EU and go back to its own currency(Drachma). This would mean heavy devaluation of the Drachma and trust on the Euro would be lifted. Euro zone would not be stable (but in reality it never was), and it could cause chain reactions to other countries in Euro Zone (as all the debt has been interlinked to each other in Euro zone)
Hence it called on referendum and urged people to vote NO. (referendum was on whether to accept bailout and continue to be part of euro zone and accept austerity measures) and Greek people voted NO.
However when Greece defaulted , Greece wanted another deal with creditors.
The option when Greek people voted NO was that government would start paying expenses with IOU's ( as secondary currency that would be worthless out of Greece) . Another option is converting all Euro debt to Drachma , which would allow Greece to pay debts with depriciated currency.
The first option is that Greece can do without leaving Euro zone.
In principle, leaving Euro Zone and devaluation of currency would mean improved trade and demand and attractive tourism. However if past is any indication , it has already tried to cut wages without much positive results.
Additionally leaving Euro Zone would mean that Greece would have to pay higher interest rates .
Whether Greek Govt. accepts bailout or leaves Euro zone is upto them.