There are actually three ways that institutions, corporations and individuals trade forex: the spot market, the forwards market and the futures market. The forex trading in the spot market always has been the largest market because it is the "underlying" real asset that the forwards and futures markets are based on. In the past, the futures market was the most popular venue for traders because it was available to individual investors for a longer period of time. However, with the advent of electronic trading and numerous forex brokers, the spot market has witnessed a huge surge in activity and now surpasses the futures market as the preferred trading market for individual investors and speculators. When people refer to the forex market, they usually are referring to the spot market. The forwards and futures markets tend to be more popular with companies that need to hedge their foreign exchange risks out to a specific date in the future.
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.