Money serves three functions: a medium of exchange; a store of value; and a unit of accounting.
Money vs currency
Currency is the money officially used by an administrative unit. There can be in-store currency (arcade tokens) and of course national currency (the United States dollar).
Fiat money derives its value from the government.
A unit of currency corresponds to a fixed unit of a commodity like gold or silver.
The gold standard is a well known form of commodity money. The silver standard has also been used historically.
Fiat money vs commodity money
Fiat money gives the government strong monetary control.