Currency exchange rates are determined everyday in large global currency exchange markets. There is no fixed value for any of the major currency -- all currency values are described in relation to another currency. The relationship between interest rates, and other domestic monetary policies, and currency exchange rates is complex, but at the core it is all about supply and demand.
Interest rates influence the return or yield on bonds. Because, for example, U.S. Treasury bonds can only be bought in U.S. dollars, a high interest rate in the U.S. will create demand for dollars in which to purchase those bonds. A low interest rate, relative to other major economies, will reduce demand for dollars, as investors move toward higher yielding investments. At least, this is true in normal periods of economic expansion.
The relationship becomes a bit inverted, however, when investors become highly risk averse. In periods of credit contraction or recession, money will tend to move into safer assets, driving down interest rates. The low yield on bonds then is a reflection of the demand for their relative safety and low credit risk, and not a deterrent. In late summer 2008, for example, the U.S. dollar gained value against the euro even as interest rates in the U.S. were significantly lower because the likelihood of a U.S. default on Treasuries was deemed less than in Europe. The lack of a federal treasury-system meant responses to bank failure would be country-specific, keeping interbank lending rates in Europe at alarmingly high levels.
Interest rates can also have economic effects, which influence currency exchange. Following the idea of supply and demand, speculators favour the currency of economies that are expanding, creating a virtual cycle of appreciation. An economy who's GDP is rising faster than its monetary base is by default increasing the value of its currency, and this will likely be reflected in currency exchanges.
Interest rates can also have an effect on foreign countries. Japan, for example, set its interest rate well below the rest of the world. The result was a carry trade where speculators borrowed from Japanese banks and converted the yen into other higher-yielding currencies, driving up their relative value in the process. Unfortunately, this effect was one of the principal causes of the global savings glut that triggered the massive global banking failures of 2008.