How Does BitCoin's Economy Work?
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.