Fiat money means all kinds of money that are made legal tender backed by a government. The term is usually reserved for legal-tender paper money or coins that have face values and not pegged against the commodity values such as gold or silver.
Historically, paper money and banknotes are used as promises to pay the bearer a specified amount of a precious metal, typically silver or gold. The US dollar served as an international reserve currency post WWII, backed by gold at a fixed value of $35 an ounce.
During the 20th century, gold price was hard to maintain at a fixed rate hence they have stopped the convertibility of the dollar into gold or other reserve assets. Most major currencies’ rising and falling in value against one another based on market demand. A US dollar can’t be traded for a set quantity of gold anymore. It can only buy a dollar’s worth of the shiny stuff, based on the value of an ounce of gold. That is what it means to have “fiat currency.” A dollar is worth a dollar –- nothing more, nothing less. The number of goods that it can buy changes based on what is happening in the market for those goods.
With fiat currency, the government can easily change the relative value of the currency by printing more of it or removing some from circulation. That gives the government far more power to influence the economy. If it needs to stimulate the economy, it can print more money. If it needs to fight off inflation, it can pull some money out of the market and put it in a vault.