I think maybe a good way to approach the issue is by first looking at direct exchange (i.e. barter).
Let’s say Smith has cows and Jones has chickens. Now for some reason, Jones would like to have at least one of Smith’s cows. He figures that Smith won’t be willing to just give him one or more of them. The only thing that Jones can offer in exchange is chickens.* So the question is, how many chickens is he willing to give up for one of Smith’s cows?
There’s no correct answer to that question. It’s all a matter of what Jones prefers. On the other hand, it’s not entirely up to Jones, for if it were, he’d be able to get one or more of Smith’s cows for free. After all, that’s surely what he prefers the most. So it’s not just a matter of preferences per se, but a matter of marginal preferences - what’s the largest number of chickens that Jones prefers one cow over?
Now what about Smith? Who’s to say that Smith will prefer to give up one of his cows for the same number of chickens that Jones is willing to give up in exchange? The answer is no one, of course. But if Smith doesn’t accept Jones’ highest offer, then there will be no exchange. Of course, if Smith doesn’t accept Jones’ highest offer, that also means that Jones doesn’t accept Smith’s highest offer.
But let’s say that Jones and Smith reach a deal. Where does the price come from? Well, the price emerges from Jones’ and Smith’s interaction. It’s an exchange ratio of cows to chickens that both men find preferable to their current situations. For example, if they agree to trade one of Smith’s cows for five of Jones’ chickens, then that means Smith prefers the five chickens over the one cow and Jones prefers the one cow over the five chickens.
One very important thing to note here is that, strictly speaking, every price is unique. Why is this? Because every exchange is (again strictly speaking) unique. So where do market prices come from? Well, if you conduct an exchange with someone in front of other people (i.e. “publicly”), you’re effectively sending a signal to those other people. This signal is “I’m willing to exchange good/service X for good/service Y at this ratio.” Once you send that signal, those who know about it will likely see no reason to offer more than that ratio. It also serves as a signal to others who are trying to conduct the same exchanges (i.e. your competitors) - they’ll see less reason (if any) to demand more than that ratio.
With the above, note that money hasn’t been mentioned at all. That’s because money isn’t strictly necessary for these economic phenomena to come about. Prices are simply exchange ratios of one good/service for another. What money does is help make exchanges happen, albeit in a less direct fashion.
- Strictly speaking, Jones can also offer to exchange his labor (time and effort) for one or more of Smith’s cows.