It depends on the currency. Sometimes currency exchange rates are "pegged" to another currency, for example a central bank declares that 1 peso = 1 dollar and that's that.
Not that's not real value, real value is based on what people are actually willing to pay for it. Most major currencies are "floating" currencies, and the market determines the exchange rate based on what people are willing to pay for it.
So at the heart of the system what determines it is people. For example, I live in Costa Rica. I primarily use dollars but also use the Costa Rican Colon. If the dollar is too high, I would not buy dollars as much. When it's low I'll buy more.
The differences in volume for these trades are the fundamental basis of determining the exchange rate. The market behavior is motivated by a variety of things, but the collective result of the behavior is the exchange rate.