From Econbrowser by James Hamilton:
Instead of having a sum (in dollars) in an account with a bank, you could have a sum (in Bitcoins) that you hold in an account that is kept track of by a network of individuals with a public record of where all the sums reside. The mechanics of being able to transfer an entry from one Bitcoin account to another are based on advances in cryptology that use open-architecture algorithms to convert one string of data into another. You can see one in operation here. You enter one string of characters, and out comes another string. Although the formulas by which the output is calculated are totally open and public, it is essentially infeasible to do the operation in reverse. If you only know the string that came out as a result of the operations, about the only way you can guess what went in is by trying every possible input string, a very time-consuming process even for the fastest computers. On the other hand, if you tell me the input you used and I already know the output, I can readily verify whether the input string was indeed as you reported.
The output of a Bitcoin transaction is based on combining some private code associated with your holdings, which only you know, with the full history of previous transactions, which everyone knows. If you supply the correct private code, other users can verify that you indeed were the owner of that sum because your code together with the public history correctly solves a known math problem. In this way, your participation is required to transfer your sum to a new owner, with security of the system maintained by the difficulty of anyone simply guessing the code.
OK, so let’s grant for the time being that the technology exists for you to securely order the transfer of some of your Bitcoin holdings to somebody else without any government or bank needing to be involved. Why does the stuff have value in the first place? The answer is that it would be very helpful to many buyers and sellers of real goods and services if they were able to pay for transactions in this way. We can think of any form of money as an asset that provides liquidity services, which refers to the tangible benefit to its holder coming from the ability of the asset to facilitate certain transactions. The value of money, that is, the value of real stuff you’d be willing to give up to hold money, can be thought of as the present value of the stream of these future liquidity services.
Bitcoin has two potential advantages over credit cards for providing such liquidity services. First, the supporting network only needs to verify that the private code is valid, which is less costly than verifying that you are indeed the rightful owner of a credit card and are ultimately going to deliver good funds. With a conventional credit card, the merchant needs to pay the card company a significant fee for the transaction which in an economic sense results from that high cost of verifying everybody’s compliance. That’s why many merchants are embracing Bitcoin. Your Bitcoin gets deposited into the account of a third party that the merchant specifies. That third party then gives the merchant dollars, confident they will be able to get dollars in turn from somebody else who will want the Bitcoins to pay for some other transaction.