The forex spot rate is determined by supply and demand. Banks all over the world are buying and selling different currencies to accommodate their customers' requirements for trade or to exchange one currency into another.
For example, an American bank receives a deposit from a German bank on behalf of their client who wants to buy something from a company in America. The German client has to pay the American supplier in dollars. The German client has euros and these euros need to be exchanged then for dollars. The German buyer will instruct his bank to exchange the euros to dollars and transfer the money to the U.S. supplier. If the bank doesn't have a supply of dollars, it will buy the dollars from another bank and sell euros.
The sum total of all banks selling dollars and all banks buying dollars creates a supply and demand for U.S. Dollars. If the demand for dollars increases then the dollar will appreciate against other currencies. If the demand drops then the dollar depreciates against the other currencies.
The rates are set by all the participating banks bidding and offering currencies all day long amongst each other. This is the interbank system and is the way currencies are traded and the way exchange rates are determined. (For more, see The Foreign Exchange Interbank Market.)
This question was answered by Selwyn Gishen.