Say, in light of current events, all member nations of the EU decide to go their separate ways, and end the Euro with it. If this occurs, what happens to the foreign exchange market on said currency?
Unwinding the Euro as a currency is extremely more complex than the UK voting itself out of the European Union. There are (were) 28 European Union member countries, 19 of them are on the Euro currency.
The Euro is a fairly new currency at about 20 years old. Read about the installation of the Euro to get an idea of what happened to Deutschmarks and other currencies as an example. Essentially each country's currency is assigned an exchange rate that's guaranteed by a central bank for a specified period of time. All holders of that currency trade their currency and you're off to the races.