I know money has value because people will accept it for payment, but what gives it its underlying value? I know it is no longer backed by gold or anything else. I know money can be used to pay taxes. Is that where all the value comes from? Is all the value created by the demand to get money to pay taxes? Is the value of money affected by tax policy?
The underlying value comes from the fact that people believe that the US government will honor its debts. This is exactly the same sort of value that’s ascribed to gold-backed currency: in that case, the value comes from the fact that people believe that the US government will honor its promise to give you gold in exchange for the proper amount of currency.
Honor what debt, though? What is the money backed by? What is the government promising?
The dollar has value because it always has, since the earliest days of the republic; because the U.S. is now the sole remaining superpower, politically, economically and militarily; because the U.S. Government has always honored its financial obligations; because the Federal Reserve is widely respected as a smart and skilled (if not infallible) decision-maker on monetary policy; and because the dollar has over time become accepted across the world as a reliable and relatively stable currency.
I understand that all of this is somewhat circular reasoning, but there you have it. And whether all of these conditions will continue indefinitely is now, unfortunately, at least somewhat in doubt.
No need to specify that for the dollar, though. All currencies have value since the Central Banks back them up; be it the Sole Superpower or the Tiny Fisherman Island in the middle of the ocean.
ETA: And now I read the thread topic properly. So I do beg pardon for the snark.
Central banks have, from time to time, gone belly up. Even in these dire economic times, however, the likelihood of the Fed doing so is about on the order of the Sun exploding.
Yeah, that’s kinda* true. I just wanted to point out that the underlying value of every currency is that the Central Banks “say” it has value.
Basically what every country promises, with its currency, is that you can buy stuff with it within its borders. And that’s the underlying value.
*I’m not up-to-date on the current odds for the sun blowing up tomorrow 
Money has value because people act as if money has value. We all agreed to use currency to represent (labor, materials, anything that is difficult to carry around) value so as to be able to trade our (labor, materials, anything that is difficult to carry around) for stuff we want.
Money by itself has no intrinsic value, but we have agreed to use it in place of the abstract concept of “value” because it makes trade easier to have standardized tokens.
But that does not set the value of money. Why Apple agree to sell an iPod for 199?
Why is a dollar worth X and a Euro Y, if they are both backed only by a promise? What is the promise?
Because people agree to buy them at 199. It’s a kind of dance. The maker of the iPod may consider a minimum price based on what it cost to make the iPod, but beyond that, it’s totally free-form. The maker can set the selling price at its making cost; can set the price as high as it wants, or can set the price lower than cost if it sees an advantage to getting units out there and is willing to subsidize their sale.
Example: at work, we had bags of Crunchy Organic Snacks[sup]TM[/sup] for $2.00 each. Very few sold. One day the cafeteria reduced the price to $0.50. Now the bags of Crunchy Organic Snacks[sup]TM[/sup] are selling. So what is their value? S2.00 or $0.50? Value is what people will pay for something.
The promise that you can use the money and that it’s worth something. It’s abstract
And the reason that the Dollar is worth X and the Euro Y is due to supply and demand. People are willing to pay X Dollars for Y Euro, just as the pre-historic you would be willing to pay for example two fish for a goat-skin.
And in the same way that if loads of people all of a sudden have fish, the price of a goat-skin would rise (since so many people have fish), the currencies fluctuate.
Make sense?
You know roughly how much you have to work to acquire $199.00. You know roughly how much food/gas/shelter/entertainment that $199.00 will buy you if you don’t spring for an iPod. So you can make the decision if the value to you of owning the iPod is greater or equal than the value of the work and the other goods you’re giving up in its stead. This all works out much better than your showing up at the doors of Apple, Inc. with a cow, a pig, and some homemade baskets and trying to barter for an iPod.
The only real understanding that you appear to be looking for will either take an internship at a major financial bank doing debt/credit risk management or currency trading, or some successive courses in Economics at the University level. The short answer has been said already, it’s because it’s what people come into agreement with. The dollar and the euro float, meaning that they don’t have set ratios against each other (or against other currencies; though, other currencies do, like the Yuan (pegged against the US dollar)).
Why is this? Again, the short answer has already been given: it’s what people believe. They believe that amongst a whole host of factors: liquidity of the currency, backing by government, value of goods and services made in that country, etc. This is decided by markets. Which ones? A bunch of them, everything from currency trading, stock exchanges, bond clearing houses (or whatever they’re called), future exchanges, etc. All these markets help determine the value of the currency.
The promise is that you can go into any bank in the US and they will accept dollars as legal tendor and that you can use those dollars to pay your taxes. The same does not apply to seashells, goats, pigs, or magic beans (although one could presumably find a market for those things).
Since that is what the banks are using, everyone else finds it convienient to also use dollars.
As to the question of why things cost the way they do, the Fed controls the money supply through monetary policy. When things start getting more expensive (inflation), they can shrink the money supply through interest rates. In a sense, money becomes more scarce and more valuable. The opposite is true as well. If the government just started printing money, the value of money decreases and inflation increases. If they start printing a shitload of money, the economy would turn into pre WWII Germany where people would burn currency because it was cheaper than using it to buy firewood.
If the abstract-but-accurate answers above don’t work for you, there’s another way to look at it:
Long ago, a central authority (e.g., the King of Mercia) might say that a coin containing a certain amount of precious metal (e.g., 1.5 grams of silver) was to be minted. This became what people traded, and it was easy to understand that a silver penny had value, because you could always melt it down and have silver. Wonderful.
What mechanism decided that a silver penny was worth, say, three sheep? The very same mechanism that decides 199 paper dollars is worth an iPod: the market. If you take your three sheep to the monthly The-Big-Local-Town-Burgh market and tried to get 5 pennies for them, you might get some-one to pay you that, but probably not. If you take your iPod to the general retail market and ask for 399 paper dollars, you will get a few people to pay that, but not many. Dollars and silver pennies both are ultimately worth whatever people are willing to trade for them.
Yet another way to look at the value of a dollar is to think of the value of everything made or produced in the USA and say one dollar is worth approximately 0.00000000000013 of that. And by “everything made or produced” I don’t just mean all the physical goods, I mean everything people will pay money for, including a landscaper’s labor, a CEO’s ideas and experience, and even internet access. This may not be a strictly accurate economic perspective, but it is a useful one.
So it comes back to demand created by the need pay taxes? If the government lowers taxes does it devalue money the same as printing more money? If there was no taxation would money have value?
Isn’t it more than “the government will accept taxes paid in it”? I thought the whole “This is legal tender” bit printed on every bill meant that by law, people have to accept money to pay off any debt?
Anyway, the real reason that money has value is because people believe that money has value. This can be disconcerting to many. But the alternative is to barter goods that have real value, a horrifically inefficient process. Consider: right now if you want a sandwich, you can go to any store and purchase it with money. In a barter economy it’s no where near that easy. If you want a sandwich and the only goods that you have in your possession is a cow, you have to find a sandwich retailer who wants a cow. And if you don’t believe that a cow is a fair trade for a single sandwich, you have to find some way for the retailer to make change for a cow.
Some advocate using gold as currency, but the value of gold is largely as imaginary as the value of fiat money. Gold is valuable largely because people believe that gold is valuable.
The promise is written right on the bill: “This note is legal tender for all debts, public and private.” Which means if you owe anybody money in the United States they have to accept the US dollar, and if they refuse the government will force them to.
The eruo has the same promise, but he reason value of one goes up in relation to the other has less to do with people’s faith in that promise and more to do with market forces like supply and demand and economic conditions like inflation. Almost no one thinks the US or the EU are going belly up and their currency won’t be honored, but like others have said just because you know the promise is good doesn’t mean one currency can purchase as much stuff as another. It just means it isn’t totally worthless.
All finances are based on belief. If you loan your buddy a hundred dollars, you’re doing it because you believe he’ll pay you back and not run off with your money.
Money is based on the same system. You accept a dollar bill as having some value because you believe that somebody else will also accept that dollar bill as having that same value.