“Forex” is short for the foreign exchange market. The market is for trading moneys. It is global yet decentralized. It determines the value for different forms of money against each other. The prime function of the foreign exchange market is conversion of currency. Speculation and carry trade both are part of the system. In the most basic form the exchanges that take place are a simple purchase of one type of currency with a equivalent amount of another type of currency.
The forex is an over the counter or OTC market, drug meaning that brokers and dealers negotiate with one another without going through a central exchange or clearing house anywhere. Although the market lacks a central exchange, drug it is anchored by a number of financial centers (New York, Tokyo, Hong Kong, Singapore and London) worldwide, the largest being London. It is because of this that they typical quotes listed for currency value are more then likely the London market price, which can be seen if one looks at the IMF values (they are based on the noon values from London).
In the 1970s various governments moved to floating exchange rates. Before this time exchange rates were fixed based on the Bretton Woods system devised after World War II. With this change came the forex. It is unique in several aspects, including a huge trading volume and high liquidity, world wide reach, the hours of operation which are 20:15 GMT Sunday through 22:00 GMT on Friday, the large number of things that impact the exchange values, the seemingly low amounts of relative profit in contrast to other types of markets, and the use of leverage to create larger profit margins.