Who determines the exchange rate between currencies of different countries?

Well it depends. If you are talking about free market economies, then the market is the one that sets the exchange rate... its pure supply and demand. Let me give you an example.

Take for instance what happened today to the exchange rate between the mexican peso and US dollar. Investors around the world are thinking that the Fed is going to hike up the intrest rates, making american investment more intresting than before. So what happens to the peso? Well, investors that have their money in the mexican stock market or in mexican currency start to convert their pesos to dollars to bring them back to the US to invest. Since more people are demanding dollar in exchange of pesos, the supply is forced to adjust prices. Basic economics, more demand for a good makes price go up... so the exchange rate pesos for dolar is affected negatively... inversly, demand of pesos for dollar goes down! This happens with all exchange rates out there.

Another example. The middle east used to quote and sell barrels of oil in us dollars. When some nations decided to use the euro instead of the dollar, the dollar for euros exchange rate went up, since less dollars were being demanded, and more euros were being demanded to buy the oil.

That is the free market exchange rate... now, there is also a theoretical echange rate that uses intrest rate from both countries to calculate the exchange rate. The actual exchange rate in the market may be diffrent from the theoretical one, showing an over or under-valuation of a currency, depending on the difference.

Currencies that come from closed economies may have a fixed exchange rate, regulated by goverment. On this examples, the currency from the not opened economy might be overvalued by goverment fixing to protect certain intrests... sooner or later, the countries reserves mught not hold, and a hard depreciation of the currency is bounded to happen.

Hope i explained myself... kind of late and im almost ready for bed. Any more questions, feel free to email me!

Good luck!

Who determines the exchange rate between currencies of different countries?

the stock market, and how the dollar fares against the euro, or whatever else you may be exchanging dollars for