Fiat money is physical money (paper money and coins), while representative money is something that represents intent to pay the money such as a check. Fiat money is backed by the government, and representative money can be backed by different things. For example, a personal check is backed by the money in a bank account. Without backing, both fiat money and representative money are completely worthless.
Today, most representative money is something backed by fiat money. In the past, the money produced by a government was considered representative money. For every quantity of money printed, there was enough gold or silver to back it. A person could actually go and exchange the money directly for gold. However, many governments fall to the lure of printing too much paper money, which leads to inflation. A dollar is no longer worth a dollar in gold. When this happens, the money becomes fiat money.
Instead of using gold as the power behind the money, the government is the strength and the reason the fiat money has value. The money has value because the government says it does. In turn, people want the fiat money. If the government fell on hard times, or if people everywhere suddenly did not want a form of currency such as the US dollar, it would lose all of its value because there is no physical gold behind it.
Most modern money is fiat money because most governments have printed too much money to compete with inflation. The U.S. dollar shifted from representative money to fiat money when President Nixon decided to abandon the gold standard in 1971.