For almost two thousand years, people used metal coins for money, or they used letters of credit (like a check today). But there was a big change around 1150 AD in China, under the Song Dynasty, when people first started to use paper money (a promise to pay gold or silver or bronze in exchange for this piece of paper). People in China had also invented the first metal coins. So people in China had the first idea for both coins and paper money.
When the Mongol Empire united Asia in the 1200s AD, the idea of paper money spread to the Islamic Empire and then to Europe. Here, too, people began to use letters of credit as a substitute for money. These were letters from a bank, that you could carry around with you when you were travelling. When you got where you were going, you could show your letter to a bank there, and that bank would give you some money. That way, you didn’t have to carry bags of money with you when you were travelling and they might get stolen. Modern checks and credit cards are based on the same idea.
Learn by doing: look at paper money from different countries
More about the invention of coins
More about the beginning of bookkeeping
Eyewitness: Money, by Joe Cribb (2000). Not the best in the series, but still a good introduction to exchange systems.
The History of Money: From Sandstone to Cyberspace, by Jack Weatherford (1998). Great on what money is, and how it has changed over time – some conclusions are controversial.