At least in the USA, Federal Reserve Notes are backed by government bonds- debt instruments based on the government’s sovereign power of taxation. That’s the difference between US dollars and Zimbabwe dollars: the latter isn’t raising (anywhere near enough) revenue by taxation, it’s simply handing out pieces of paper.
What’s an IOU for $100 worth to you?
If you trust the person and know they will soon have the money and pay you back it’s worth $100. They might even throw in something extra for your trouble.
From some other random slacker it may be worthless.
The issue of trust is the main building block of a healthy economy. And not just the currency. You trust a bank to return your deposits. You trust the local officials to not walk in confiscate your stuff.* Etc.
So we trust that when we get money from someone that someone else will accept the money in payment later on.
Too many people either don’t understand the role of trust or have a problem with trust at such a large scale involving so many people. But group trust is apparently an intrinsic aspect of the nature of most people.
- Which is why on-the-spot civil forfeiture is such a concern to so many people.
I think you overcomplicated the example by imagining something with a price tag on it and then the shopkeeper refusing that number of dollars. Yes there are laws about this, because it also ties in to people’s ability to settle debts (if you owe me $100, I can’t refuse a $100 bill as payment, and take action as if you are still in arrears).
A simpler example might be something like: I have a golden fleece, and I am only willing to hand it over if you answer me my riddles three. The government can’t force me to exchange it for currency.
Because, for one thing, what number of dollars would I be told to accept?
If you think gold has any intrinsic value, imagine a post nuclear situation and you are in a shelter that has a year’s supply of food and someone came by and offered you a pound–or a ton–of gold if you would leave your shelter to him. Would you take it? Food and, to a lesser extent, shelter and clothing have intrinsic value. Very little else.
FWIW, some nations or people within those nations have switched their currencies when their national currency lost all value.
The people in a nation switching to Euros or USD when their own currency has issues happens too.
Regarding gold having value because of being worn in jewellery:
It’s partly as a display of wealth, so it is acting like a fiat currency even there.
Aluminium is plenty shiny, and was a much-prized jewellery back when it was scarce. So why don’t rappers wear big aluminium chains?
It’s because a significant point of jewellery is to flaunt excess wealth. Aluminium is too common for that; it’s like a hyperinflated currency.
I agree, but gold still has value to a person who doesn’t wear jewelry. Maybe I think gold doesn’t look all that great, but it still has value to me because I can sell it to others for money which I can then use for things that do have value to me, say a new TV set.
I think the idea that gold has value because it is shiny and pretty, much like the idea that money has value because it can pay taxes is a narrow view of the concept.
The same reasoning can be applied to diamonds. They are valued because they are a symbol of wealth. In truth, there are so many diamonds existing that they are practically worthless.
This may be more that food and shelter are now much more expensive. 
Your boss pays you $X in wages. You buy $Y worth of groceries. You are trading labor for food. The dollar is just a bookkeeping gimmick.
Your landlord collects $X in rent. He pays $Y for chemotherapy. He is trading lodging for medicine. The dollar is just a bookkeeping gimmick.
You can keep track of the exchange rates between labor and food, or labor and lodging, or lodging and medicine, or medicine and labor, or between any pair of goods or services you want. It’s easier just to calculate everything with one yardstick. You and your fellow citizens can pick any yardstick you want. Silver-backed Mexican pesos. Gold-backed Spanish pesos. Gold-backed U.S. dollars. Fiat dollars.
When the government mandates a change from gold-backed currency to fiat currency, it is just changing the yardstick from English to metric. The values of the other commodities relative to each other have not changed, and eventually a new equilibrium will emerge. (Usually not the one the politicians wanted.)
When the government mandates an exchange rate ($35 for an ounce of gold, $15 for an hour of manual labor, $0.00 for an appendectomy, etc.), it is just changing the yardstick from English to metric. Values of other commodities relative to each other have not changed. The new measuring standard will work its way through the economy until a new equilibrium emerges. (Almost never the one the politicians wanted.)
Money has value for exactly the same reason that everyone driving on the same side of the road has value.
If you can understand why you, personally, would like to drive on the right side of the road when everyone else drives on the right side – and likewise drive on the left if everyone else is driving on the left – then you can and do understand why money is valuable.
It is the same underlying mechanism.
We constantly have need to coordinate our actions with the actions of others. Essentially nothing in your home was produced by you, alone. You work together with others to produce the value you see around you.
The multitude of people who help make your life a little bit better live, quite literally, all across the globe. One of them in one country makes your life a fraction of a penny better here, another in a different country makes your life better a fraction of a penny there. There are billions of us. Those fractions of pennies add up. If the people offering those bits of value had to help you directly, if they had to show up at your doorstep to do you a solid, at exactly the same time that you did them a solid, so that they and you exchanged your valuable work directly, and in person, then that would be ridiculous.
It simply would not work.
So by collective agreement – a mechanism people can easily understand when we think about the value of driving on one side of the road – we earn our value indirectly. I teach economics. But I don’t teach it to the people at the grocery store in exchange for groceries. I teach it to people who want to learn it, in exchange for tokens of value that I know people will accept, and then I trade those tokens of value to the grocery store. And then the people at the store can use the same tokens they received from many for something they find more valuable than economics instruction.
This works because we all choose to intermediate our value using the same tool. We use a common medium of exchange across large regions. A medium of exchange is valuable precisely because everyone else uses it, just as driving on the right side of the road is useful precisely because everyone else does, too. When I lived in Japan, I used yen for my transactions just as I drove on the left. I used the same tool everyone else used, because coordination is how we help each other build value. We coordinate around a common money because that’s useful. Simple as that.
Money can be complicated.
But why money is valuable is not complicated. It’s valuable because we all coordinate to make it valuable. The formal name for this is a Nash equilibrium. We could, potentially, coordinate an equilibrium of monetary value around anything, but of course existing governments can and do strongly guide the mechanism of coordination.
This is insufficient.
Bitcoin is closer to digital commodity than digital currency, but it’s close enough to demonstrate a few important principles. It obviously has no intrinsic value, since it’s just digits on computer drives. But it is clearly valuable: it is bought and sold in exchange for other valuable things. Even so, no government accepts it for taxes. Its value is sustained by some other mechanism.
Government taxation and regulation can and does guide a society toward a particular item around which we coordinate our collective exchange of value. Such coordination is one of the primary purposes of government. But government can’t be strictly necessary to sustain it, because we see people negotiating value of otherwise apparently worthless objects even without imposition of that government-sponsored coordination.
The only thing that’s necessary for currency is the conviction (on the part of a reasonable number of people) that you will eventually be able to trade that currency for something of value.
You don’t have to believe the currency has any value in and of itself–you don’t have to believe that anyone else believes that, either. Just that there is someone that will take your simoleons in exchange for food or something else.
As noted, cryptocurrencies are the ultimate expression of this. They aren’t backed by anything; there’s no central authority willing to take them for tax or anything else. Just a bunch of people who think that, if someone gives them Bitcoin for one thing, they’ll be able to trade that for some other thing later.
A consequence of this is that a currency has to be scarce. A currency that can be easily duplicated is worthless. So either you have an authority enforcing the scarcity (limited minting, anti-counterfeiting), or it’s a natural effect (only so much gold), or it’s somehow designed into the currency (cryptocurrency). That’s not a sufficient condition, but it is a necessary one.
There was a story on the radio about a couple travelling around the world on a yacht. At one remote Pacific island (they say) they wanted to buy some meat for provisions, and offered to pay for it with dollars or Euros. The locals didn’t want to deal with it–there wasn’t any supermarket on the island where people could easily spend any of that cash to replace the food. They ended up bartering meat for cheese.
ETA: there are islands where getting your hands on some food, or clothes, or even fresh water, is a little more complicated than heading to the bodega around the corner.
Where is this cache? Fort Knox?
I’m pretty sure I’ve told this story here before, but…
When I was a kid, every Halloween, my extended circle of friends (about 15 of us) would converge on the home of the kid who lived in the best neighborhood. We’d all use that as our home base for trick-or-treating, and then at the end of the night, sit in a big circle in the living room trading our candy. A wonderfully chaotic scene, as I’m sure you can imagine.
Well, it came to pass that everyone was willing to trade other candies for Good-N-Plenty. How did it come to pass? I don’t know, because none of us actually liked Good-N-Plenty. But we would all trade other candies for it, because we all knew that someone else would then trade us other candy for it. In short, we turned Good-N-Plenty into a currency, with no fixed backing, no governmental coordination, and no inherent value. And amazingly, it worked.
I would have been a sticking point in that economy: I like black licorice, so I would have taken a few boxes of Good-N-Plenty out of circulation to eat them, exerting a deflationary pressure. In short, I would have bid up the price of a box of Good-N-Plenty by making it so there were fewer boxes of the stuff chasing all of the rest of the goods. I might have even incentivized others to trade me Good-N-Plentys by offering above-market prices for them, like a fun-sized Snickers or something. If that goes on too long, people can’t use them for trade anymore, because too few boxes exist for an economy of them. In longer-term scenarios, the money gets hoarded because hoarding has a better guaranteed rate of return than investment, which is fatal to the economy.
Compare this to Ancient Egypt, which had a gold-based economy and a religion which demanded they bury gold in tombs every so often. Take that to the infinite limit and you have an economy where all the gold is entombed and not, in fact, in the economy. Of course, that doesn’t account for grave robbers taking gold out of the tombs and putting it back into the economy.
There’s a common lesson here: Don’t have money people might want to do other things with, like eating or burying under the ground. Keep your money something which only has exchange value, so it can continue to do its most important job, circulating through the economy.
This is why we need to outlaw billionaires (or tax them into destitution) because they just want to use their $100 bills for lighting their cigars. 
As mentioned upthread, money is a bookkeeping trick. I don’t want to barter lumber from my wooded property or my obsolete COBOL skills for deli sandwiches or a hooker. Too clumsy. So we set “prices” and finesse the exchange of goods and services. The “value” of any good or service is whatever the parties agree on. Money is thus worth whatever you did to get it and whatever you can swap it for.
These examples of Good-N-Plenty or other specie money increasing in value reminded me of a famous case where specie gold decreased in value in Egypt and its neighbors. I refer of course to Mansa Musa’s famous Hajj 700 years ago.
If the numbers are to be believed, I think Musa started with about a million troy ounces of gold in his caravan.
What happened to Spain in the 17th century as galleons brought a flood of gold and silver from the New World? I hope **Hellestal **or other experts can comment on this.
I think this illustrates very well why money backed by a commodity, although working well for an ancient civilization, does not work well when it becomes more advanced.
Your story would be much the same as the effect on the value of Microsoft stock if Bill Gates decided to sell all of his (or indeed give it away to keep with part of the parallel).
Fiat currency, as long as the fiat part of it is used responsibly and honestly, is much better than currency backed by any commodity. And if you cannot trust the government to be responsible and honest with fiat currency, then why would you trust it to honor its pledge to back its currency with a commodity?
“Money” as a store of value is pretty basic in civilization. In prisons where guys are denied money as a method of trade, they use cigarettes (where still allowed) or items from the commissary such as chips, candy, sodas, or other sundries as a medium of exchange. I might hate Ramen Noodles, but I know I can trade it for toothpaste so I accept Ramen Noodles as payment. To understand money is just to elevate this understanding up a few notches.
ETA: Ask yourself, why would a prisoner who does not smoke accept cigarettes as payment?