What is fiat currency?
Money that works because everyone agrees it does.
A fiat currency is money that has no intrinsic value and is not redeemable for any commodity. A ten-pound note is a piece of polymer worth a fraction of a penny as material. It buys ten pounds of goods because the UK government designates it legal tender and because everyone else accepts it on the same terms. Fiat is Latin for "let it be done", the value exists by decree and by convention, not by substance.
What it replaced
Under a commodity standard, a note was a claim cheque. You could in principle hand it to the issuing bank and receive a fixed weight of metal. The note's value was tethered to that promise, which in turn tethered how much money a government could issue to how much metal it held.
That tether was cut in stages through the twentieth century, ending with the United States closing the gold convertibility of the dollar in 1971. Since then essentially every circulating national currency has been fiat.
What actually backs it
Not nothing: but not a commodity either. A fiat currency's value rests on the credibility of the institution issuing it: whether the central bank keeps inflation under control, whether the state can raise taxes payable in that currency, and whether people continue to accept it from one another. These are real constraints, which is why fiat currencies mostly hold value and why the ones that collapse do so when those constraints fail.
Why it matters for physical notes
Because a note's value is a claim on the issuer rather than on metal, what happens when the issuer withdraws it matters enormously. A withdrawn banknote is not automatically worthless: many central banks redeem old notes indefinitely. But some set a deadline, after which the claim simply expires and the note becomes a collectible rather than money.
The German mark can still be exchanged for euro with no deadline at all. The Spanish peseta could not, after 30 June 2021. Same idea, opposite outcome, purely because of what each central bank decided.