Forex is short for Foreign Exchange, sometimes further abbreviated as FX. It's the worldwide market where parties can exchange one country's (or union's) currency for another such as a Euro for a US Dollar and vice versa.
The simplest comparison for most beginners to understand the concept is to think of each currency as an asset that can be bought and sold... using another currency. For instance, you could buy Euros using US Dollars based on the current market value of the Euro as priced in terms of US Dollars. In that sense, you could think of the Euro as an equivalent to, say, a stock or futures contract that you can buy or sell for profit or loss.
In reality, the FX market's daily turnover volume is actually in the trillions of dollars, dwarfing the trading volume of even the world's largest stock exchanges.