Forex (foreign exchange) is the market where currencies are traded. When you buy EUR/USD, you buy euros by selling dollars — always in pairs.
It is the largest market in the world: roughly $7.5 trillion changes hands every day. No central venue — it lives across a network of computers between banks, brokers and hedge funds, 24/5 from Sunday night to Friday night.
You never physically own the currencies: you speculate on the price gap. Buying EUR/USD means betting the euro will rise against the dollar.