Contrary to the posters above, it is always the market the sets the value.

Historically, during gold/silver standard, currencies were just a title to some weight of gold or silver. The value of currency was linked to commodities value.

Currently, when currencies are not backed by any commodity, it is also the demand and supply of money that sets the price. The supply of money is to some extent (but not completely) controlled (one may say "manipulated") by the central bank, by the demand is not and the market entities decide whether it is worth to hold the currency given this price or not based on interest rates, central bank/government trustworthiness, economy situation, etc. Sometimes central banks try to fix the value of currency (or rather fix it relative to something else). That can be successful if the supply side of the money is carefully controlled. But if the market value would have been different, quickly a black market will appear with a different price.