I've seen the term "fiat currency" in another question. What is a fiat currency? Are there other types of currency?
A "fiat" currency is non-convertible paper currency that a government establishes as legal tender. Most countries today are using fiat currencies. The rest have currencies pegged (or convertible to) US Dollars (which is a fiat currency).
In the past, money was usually based on precious metals such as gold or silver. Until the end of the gold standard, you could theoretically go the the US Treasury with a US Note or Federal Reserve Note and convert the note into a fixed quantity of gold or silver (depending on the note). The US had a bi-metallic currency policy for political reasons, which means that money was backed by both gold and silver.
In short. A fiat currency is money that has value only because (usually) a government says it does.
A counter example (non-fiat currency) is a gold coin that has intrinsic value usually because it is made of valuable materials that people would trade goods/services. That is the value comes from what you are holding more than what it represents.
"fiat" in Latin means "let it be" or "let it happen" - as in "fiat lux!" meaning "let there be light!". Thus, fiat money means money that are created by the government decree - they would be random worthless pieces of paper otherwise, so the government says "let this paper be worth $100" and it becomes $100. Non-fiat money have some value that is beyond declaration - e.g., gold coin has the value of gold that is made of. Since 1971, almost all world currencies are fiat money.
There's two types of categories at play that define currency types - but I think the first is more like what you are after.
The first is there are essentially three currency types now recognised - see them described here: https://web.archive.org/web/20071020054733/http://finance.mapsofworld.com/money/types/
- Commodity Money
- Fiat Money
- Fiduciary Money
The second is currencies can be categorised by the type of economy from which they are generated (reserve/commodity/etc) - see them described here: http://www.forextraders.com/learn-forex-trading-course/major-currency-pairs.html