Forex is a commonly used abbreviation for "foreign exchange," and it is typically used to describe trading in the foreign exchange market by investors and speculators.
For example, imagine a situation where the U.S. dollar is expected to weaken in value relative to the euro. A forex trader in this situation will sell dollars and buy euros. If the euro strengthens, the purchasing power to buy dollars has now increased. The trader can now buy back more dollars than they had to begin with, making a profit.
This is similar to stock trading. A stock trader will buy a stock if they think its price will rise in the future and sell a stock if they think its price will fall in the future. Similarly, a forex trader will buy a currency pair if they expect its exchange rate will rise in the future and sell a currency pair if they expect its exchange rate will fall in the future.
The foreign exchange market is a global decentralized marketplace that determines the relative values of different currencies. Unlike other markets, there is no centralized depository or exchange where transactions are conducted. Instead, these transactions are conducted by several market participants in several locations. It is rare that any two currencies will be identical to one another in value, and it's also rare that any two currencies will maintain the same relative value for more than a short period of time. In forex, the exchange rate between two currencies constantly changes.
Currencies trade on an open market, just like stocks, bonds, computers, cars, and many other goods and services. A currency's value fluctuates as its supply and demand fluctuates, just like anything else.
An increase in supply or a decrease in demand for a currency can cause the value of that currency to fall. A decrease in the supply or an increase in demand for a currency can cause the value of that currency to rise.
A big benefit to forex trading is that you can buy or sell any currency pair, at any time subject to available liquidity. So if you think the Eurozone is going to break apart, you can sell the euro and buy the dollar (sell EUR/USD). If you think the price of gold is going to go up, based on historical correlation patterns you can buy the Australian dollar and sell the U.S. dollar (buy AUD/USD).
This also means that there really is no such thing as a "bear market," in the traditional sense. You can make (or lose) money when the market is trending up or down.