I would say that there are three (or perhaps 2) stages in the evolution of money.
The first would be barter. This is the one that you might not want to call "money" because there is no money involved. In barter, people directly exchange one commodity for another (I give you a sheep, you give me some grain.)
The second would be "commodity money." In this case, we use something as money because it has an intrinsic value. For example, we use gold coins because gold is valuable in and of itself.
The third stage would be "fiat money." This is money that has no value of its own -- we only think it is worth something because the government says so. Money today fits that category -- a $100 bill has no value because you can't really use it for anything. It's just a piece of cloth/paper and ink. But because the US government says it's worth a lot, we accept it.
Commercial exchanges existed long before the invention of money. A farmer might trade olive oil with a potter in exchange for jars, or a herder might trade wool or beast hides with a cobbler in exchange for shoes. While this system of barter still exists in many agricultural societies, and is becoming increasingly popular again with the internet making it easier to find particular exchanges, in complex societies it doesn't work out very well. If a cattle ranger needs barbed wire, he won't necessarily be able to find a barbed-wire manufacturer in need of beef; instead, he might find an accountant who needs beef, who works for a doctor who attends to the medical needs of the barbed wire company owner. These sorts of long chains quickly become unwieldy.
The next stage in the evolution of money is using some sort of generally agreed upon valuable substance as a standard currency. The earliest known such standard was cattle, which are still treated as units of value in some African tribes. Later, more portable good such as cowrie shells, gold, and silver became standard, with states or cities beginning to mint coinage.
In the modern period, financial instruments have proliferated, now including not just physical money, but virtual currencies such as the bitcoin or currencies in massive online video games. Perhaps the most dramatic moment in the evolution of modern currency was 1933 when the US went off the gold standard. Before that, currency in most countries stood for physical reserves of gold, but after that date, currencies became increasingly a matter of fiat, valued by trust and contract.