(page two; click here for page one)
Paper money (a promise to pay gold or silver or bronze in exchange for this piece of paper) was first used around 1150 AD, in China during the Sung Dynasty. 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 letters of credit and 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.
Eyewitness: Money, by Joe Cribb (2000). Not the best in the series, but still a good introduction to exchange systems for kids.
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.