What determines exchange rates?

Until 1973 the major trading countries operated under an agreement that fixed the exchange rate of their currencies relative to one another. Rates were indeed changed. but the changes were major events that surely required international agreement and came at infrequent intervals.

Given the rapid postwar recovery and increasing competitiveness of Japan and Europe, the exchange rate of the US Dollar was getting more and more out of line, it was too high, and the US trade was beginning to suffer.

In 1971, President Nixon more or less unilaterally devalued the dollar by 10 percent. And in 1973, exchange rate were set free to float, that is to be determine by market forces.

What determines the rate at which dollars are exchanged for Euros or yen on the market? We can say demand and supply, but that answer just pushes the question further back-what determines the demand and supply of various currencies?