How Does BitCoin's Economy Work? I just heard about BitCoin for the first time today, and I'm confused. 
My understanding of economics may not be the best, but I do know that money, as in the paper and coins we use in everyday life, has no value unless it's backed by something which DOES have value (such as the gold in the Federal Treasury). So I don't understand how BitCoins gain value. I understand they're created by computers doing complex mathematical equations, but...is the solution to these equations valuable? Is that what causes BitCoins to be worth something?
Hopefully the answer to my first question will answer the second as well...once BitCoins are created, how is their ongoing value determined? For instance, I heard about a BitCoin value crash today. What criteria is used to determine changes in the market, and how are exchange rates calculated?
Thanks in advance to anyone kind enough to answer my questions...I hope I don't sound like a complete idiot.
Things are worth what people are willing to trade for them. The same is true for Bitcoin. Its value is insured by nothing other then what people will trade for it. 

There are some distinctions between  bitcoin and other currencies which are worth noting. 


Bitcoin has no intrinsic value. While this is true of many modern currencies such as the U.S dollar, it has not always been the case, such as when the U.S. was still on the gold standard. (Gold has some intrinsic value such as its use in circuitry ) 
New Bitcoins get created by an algorithm which closely follows a geometric series as opposed to a central authority, such as the Federal Reserve. 


This second point while having some benefit has proved to be significant shortcoming for Bitcoin. In most currencies, as the volume and rate of transactions increases, so to does the amount of currency. This is not the case for bitcoin, and it has led to severe  hyperinflation. This is a bad thing for a currency, because if you use it as your yardstick of value (which currencies are) of other goods and services, then everything else's value is constantly being depreciated.

In short, Bitcoins economy dosent work all that well. Its based on speculation, and its only (as far as I can tell) real value as a medium of exchange comes from the drug market silk road, which uses it exclusively.