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Paper money or coins of little or no intrinsic value in themselves and not convertible into gold or silver, but made legal tender by fiat (order) of the government. 
Fiat money is an intrinsically worthless object, such as paper money, that is deemed to be money by law. To place it into historical context, one could think of three phases concerning the development of money.
First, money itself was a valuable object, such as gold, fur or peppercorns.
Second, paper money circulated, but this money was backed up by holdings of gold, and indeed could be converted into gold at a fixed price at any time.
Third, paper money circulated, but it was not backed up by anything other than the government’s promise that it will refrain from printing too much money so as to make it worthless. Since Bretton Woods, almost all paper money is of this type.
The problem, of course, was that sometimes governments broke their promise. Following World War I, Germany printed so much money that what could be purchased for one mark in 1918 cost about one trillion marks in 1923.
For instance, in 2009, one US dollar brought you more than 6 billion Zimbabwe dollars at the official exchange rate (and this was even after multiple currency reforms due to prior excessive inflation rates).
In both cases the government lost credibility so that it could not borrow to finance large budget deficits, and hence had to pay its bills with massive amounts of newly-printed fiat money.