When you trade forex, you're effectively borrowing the first currency in the pair to buy or sell the second currency. With a US$5-trillion-a-day market, the liquidity is so deep that liquidity providers—the big banks, basically—allow you to trade with leverage. To trade with leverage, you simply set aside the required margin for your trade size. If you're trading 200:1 leverage, for example, you can trade $2,000 in the market while only setting aside $10 in margin in your trading account. For 50:1 leverage, the same trade size would still only require about £40 in margin. This gives you much more exposure, while keeping your capital investment down.
Either way you don’t have to provide the full currency value to open your position. Instead you put down a margin deposit, which is a fraction of the full value. And you don’t actually buy or sell any currency: you are opening a speculative position on the change in value of the forex pair. Your profit or loss is realised when you close your position by selling or buying.
The foreign exchange market is the "place" 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.