The foreign exchange market works through financial institutions and operates on several levels. Behind the scenes, banks turn to a smaller number of financial firms known as "dealers", who are involved in large quantities of foreign exchange trading. Most foreign exchange dealers are banks, so this behind-the-scenes market is sometimes called the "interbank market" (although a few insurance companies and other kinds of financial firms are involved). Trades between foreign exchange dealers can be very large, involving hundreds of millions of dollars. Because of the sovereignty issue when involving two currencies, Forex has little (if any) supervisory entity regulating its actions.
Trading in the euro has grown considerably since the currency's creation in January 1999, and how long the foreign exchange market will remain dollar-centered is open to debate. Until recently, trading the euro versus a non-European currency ZZZ would have usually involved two trades: EURUSD and USDZZZ. The exception to this is EURJPY, which is an established traded currency pair in the interbank spot market.
The foreign exchange market is where currencies are traded. Currencies are important to most people around the world, whether they realize it or not, because currencies need to be exchanged in order to conduct foreign trade and business. If you are living in the U.S. and want to buy cheese from France, either you or the company that you buy the cheese from has to pay the French for the cheese in euros (EUR). This means that the U.S. importer would have to exchange the equivalent value of U.S. dollars (USD) into euros. The same goes for traveling. A French tourist in Egypt can't pay in euros to see the pyramids because it's not the locally accepted currency. As such, the tourist has to exchange the euros for the local currency, in this case the Egyptian pound, at the current exchange rate.