In short: Fiat money is money that is not tied to a metal; its value rests on the law, on taxes and on trust in the central bank that issues it.
Simply put: The word comes from the Latin fiat — "let it be". Since 1971–1973 the major currencies have been fiat. Three things stand behind them: legal tender status, taxes that are paid in that currency, and a central bank that keeps inflation under control (the ECB's target is 2%). This gives flexibility in a crisis, but if the institution loses trust, there is nothing material underneath.
Example: A €50 banknote cannot be exchanged for gold. A shop accepts it because the euro is legal tender and because the shop pays its own taxes in euros.
Watch out: Myth: "Money today is worth nothing because there is no gold behind it." Taxes, laws and the economy stand behind it — its value is real as long as the institution is strong, which is exactly why we should keep an eye on it.
Source: ECB, "What is money?" — https://www.ecb.europa.eu/ecb-and-you/explainers/tell-me-more/html/what_is_money.en.html · ECB, monetary policy strategy (2% target) — https://www.ecb.europa.eu/mopo/strategy/html/index.en.html