What is money? Money is the most marketable commodity. You trade it to buy things. How did we get to paper money? Well, in the past, people bought thing with gold, because it was hard money (hard money means it is hard to find, or make). However, gold is very heavy and far too valuable to use for every-day modern-day purchases. So the government decided to use paper money instead. You could trade gold for paper money, and when you wanted to buy something, you go to the bank and trade in the paper money for gold. And paper money made sense because paper money was easily divisible to use to but things and it was portable. It was not long before people figured out that the paper money also had value and started to use paper money to buy things instead of gold. The government could not print more money without mining more gold. It worked great! however, the government became greedy and started printing more money without mining more gold. This caused inflation, prices going up with the value of paper money going down.
Counterfeiting is when people illegally print money. Can governments counterfeit? Yes they can, and they do. For example, the Federal Reserve. The Federal Reserve controls how much money is printed in the US every year. Technically, paper money has no value, so all money is fake. When banks create money out of nothing (a.k.a. paper and ink),and pump it into the system, that is counterfeiting.