I like a lot of what other people have said here (faith backed by a trusted party, representation of debt, etc), but I just want to drive home the point that money has value because we said it has value. Under a system of barter, you end up with people valuing things differently. So we switched to the gold standard, but then that constrained growth. Now we have fiat currency which is based on faith.

The reason why we chose gold was because it was relatively useless at the time, only good for making things look pretty essentially. It was valuable because it was rare, and since it was rare, a central authority could control the supply because it takes a lot of capital to extract and process. Modern fiat is similar, but the rarity (making it a good substitute for value of other goods and services) comes from the government being the only person who can issue it. It's honestly kind of a weird paradox. It has to be cheap and ubiquitous enough that the supply isn't limited, but rare enough that we accept it as a stand in for value. In an alternate universe, we could have chosen river rocks (not useful for other purposes, so no one would be tempted to take supply out of the system to use for other means, and pretty ubiquitous), but we couldn't effectively control the supply.