Forex is the common name for the Foreign Exchange market, an international network of currency trading that is active 24/7.
Forex is by far the most active and highest-volume market in the world, because it involves large trades between international institutions in an effort to diversify or consolidate their exposure to various currencies. Individual traders can also participate, usually by trading nano-lots, which are 100-unit increments of currency.
The Forex market is also called the Interbank Market, because the majority of the trades are between banks. This includes the central banks of every country. In recent history, Forex trading has increased dramatically with the regulatory changes in international banking policy which opened the doors to increased liquidity between central banks and other financial institutions around the world.
These institutions seek to hedge their exposure to various currencies, since these fluctuations can be among the most troublesome for banks to manage, especially in areas of significant international trade. It is also worth remembering that Forex is an OTC (over-the-counter) market, and it lacks centralized regulation and oversight for the most part.
There will not always be transparency, and trades may occur in a take-it-or-leave-it way, but this may also be conducive to healthy competition. There are a few types of familiar trades which take place on Forex markets, such as forwards, swaps, options, shorts, and spot trading.
Spot trading just means trading at the current spot price, much how stock trading is done. Forex trading is some of the only trading that can be done 24/7.