Commodity money is money that would have value even if it were not being used as money. (This is usually referred to as having intrinsic value.) Many people cite gold as an example of commodity money, since they assert that gold has intrinsic value aside from its monetary properties. While this is true to some degree- gold does in fact have a number of uses, it's worth noting that the most often-cited uses of gold are for making money and jewelry rather than for making non-ornamental items.
Commodity-backed money is a slight variation on commodity money. While commodity money uses the commodity itself as currency directly, commodity-backed money is money that can be exchanged on demand for a specific commodity. The gold standard is a good example of the use of commodity-backed money- under the gold standard, people were not literally carrying around gold as cash and trading gold directly for goods and services, but the system worked such that currency holders could trade in their currency for a specified amount of gold.
Fiat money is money that has no intrinsic value but that has value as money because a government decreed that it has value for that purpose. While somewhat unintuitive, a monetary system using fiat money is certainly feasible and is, in fact, used by most countries today. Fiat money is possible because the three functions of money- a medium of exchange, a unit of account, an a store of value- are fulfilled as long as all people in a society acknowledge that the fiat money is a valid form of currency.
Much political discussion centers around the issue of commodity (or, more precisely, commodity-backed) money versus fiat money, but, in reality, the distinction between the two isn't quite as large as people seem to think, for two reasons. First, one objection to fiat money is the lack of intrinsic value, and opponents of fiat money often claim that a system using fiat money is inherently fragile because fiat money doesn't have a non-money value. While this is a valid concern, one must then wonder how a monetary system backed by gold is significantly different. Given that only a small fraction of the world's gold supply is used for non-ornamental properties, isn't it the case that gold has value mostly because people believe it has value, much like fiat money?
Second, opponents of fiat money claim that the ability for a government to print money without having to back it up with a specific commodity is potentially dangerous. This is also a valid concern to some degree, but one that is not entirely prevented by a commodity-backed money system, since it's certainly possible for the government to harvest more of the commodity in order to generate more money or to revalue the currency by changing its trade-in value.