A currency is a medium of exchange, that is a good B that person X trades his good A for to person Y to buy good C with from person Z since there are far more of person Y in society who want good A than there are of person Z with such a desire. For example, person A might be a baker and C a tailor. It's hard to find a tailor who wants bread, but it's easier to find someone who wants bread, trade with him for gold, then give the gold to the tailor. Historically, many things have been used as currencies – grain in ancient agricultural societies, gold later on, then dollars backed by gold, then dollars backed by nothing, then electronic records backed by dollars – and, finally, with Bitcoin, electronic records backed by nothing.
Ancient currencies, like grain, are useful in themselves – they're just used as currencies because lots of people want them, so they have dual utility as currencies and commodities. The problem with this approach is twofold. First, the optimal good that could be used as money, possessing characteristics like durability, divisibility and portability, is usually not wanted by many people for consumption. Hence, a commodity currency is generally suboptimal, forcing farmers to carry around grain when carrying gold. Second, money is hoarded. People want to reserve money on them to buy whatever goods they're planning to buy, and probably some leftover just in case. So even if person X goes to the market, sells his bread and immediately turns around and buys a shirt from person Z, there is some duration of time during which person X holds the currency. As Ludwig von Mises puts it, "Every piece of money is owned by one of the members of the market economy. The transfer of money from the control of one actor into that of another is temporally immediate and continuous. There is no fraction of time in between in which the money is not a part of an individual?s or a firm?s cash holding, but just in 'circulation.'". This, however, means that the demand for the commodity currency is higher than just the demand for the commodity, and some of the commodity that could be stored as capital, allowing people to temporarily consume more than they get because they want to undertake some productive long-term project, is instead stored as cash. Thus, there is a social loss from using such a currency, although it is justified by the social gain of more efficient trade. So from there, we moved on to currencies that have comparatively little or no value as commodities – first gold, then dollars, now Bitcoin.
But what compels our person Z to accept gold, or dollars, or Bitcoin? The answer is that there is someone else with services that person Z needs who accepts the currency. This seems like a circular argument – the currency has to circulate. If you go far enough, the whole thing becomes something like "Z accepts Bitcoin because A accepts Bitcoin because B accepts Bitcoin because … because Y accepts Bitcoin because Z accepts Bitcoin". But the argument isn't circular; it's a spiral argument: Z accepts Bitcoin because he expects that A will accept Bitcoin soon, and predictions of the future are based on historical data, so this expectation is based on the fact that A accepted Bitcoin in the recent past. Thus, we have the collapsed version: Z accepts Bitcoin at time t because A accepted Bitcoin at time t-1 because … Z accepted Bitcoin at t-26. There are thus two stable equilibrium: (1) Bitcoin has no widely accepted value, and anyone who accepts Bitcoin is simply running a welfare program for miners, and (2) Bitcoin has a widely accepted value. Thus, we can say that the acceptance of Bitcoin's value is a sort of virus maintaining itself in the collective society – essentially, Bitcoin is a meme.
The best good that can be used as currency is electronic records since they are the easiest to transfer and store and have no commodity value. But good is generally thought of as a tangible object that you own, not an electronic record linking a number to your name (whether your physical identity or something backed by a password that only you know) stored by somebody else. But electronic records possess all the characteristics of a commodity: they are scarce since the system is not willing to assign as much as you want to you, they are rivalrous, since the system enforces it that way, and they are divisible – in fact, nearly infinitely so. The idea of owning something stored by other people seems psychologically problematic. Still, we are already used to it: one can speak of "your reputation" when your reputation is, in fact, the sum of thoughts about you stored in other people's heads. Such concepts can be thought of as "social capital", just like a factory is a physical capital and knowledge in one's head is intellectual capital (not to be confused with "intellectual property", the idea of a legal property right to market share on physical expressions of certain types of intellectual capital) – they are value stored not in goods, but connections. Social capital in the pre-Bitcoin sense is a legitimate form of capital – web companies sell for hundreds of millions of dollars not because of their physical ownership of computer equipment, or even their hidden internal software, but because of the established connections between a group of employees and millions of customers. Such large webs of connections allow companies to serve millions of people rather than a few dozen for the same price, just like a factory massively increases the productive output of its workers. Electronic currencies show how physical capital and social capital are worthy of distinction for analytical purposes, in reality not so far apart. One anthropologist argues that currency was developed not from the exchange but as a formalized form of debt, which before was simply a type of informal social relation. If you have a bitcoin, what that means is that there is an agreement among its users that the world owes a certain amount of labour to you. The value of the bitcoin approximately depends on the size of the network and essentially gives you a 1 / 21000000 share in the value that can come out of this network and its ability to facilitate trade. When gold was the dominant currency, it represented physical value, while Bitcoin is a looking glass showing us what money is: social value.
But there are different types of electronic records – they can be centralized or decentralized. Centralized electronic records, such as Linden dollars as a currency, require trust in two things: trust that people will continue to accept the currency and trust that the centralized record will not fail or betray its purpose for personal gain. Like the first is a prediction of the future based on past performance, so is the second. A centralized currency is thus also backed by a reputation. A decentralized electronic record, however, has a simpler, and thus stronger, backing. Since decentralized electronic records must be designed in a way that makes them unbetrayable, thus, they are based on the trust that people will continue to use them and the trust that the network will not fail, which, aside from the unlikely event of a technical error, can only happen if people stop using them. A = B, so the two conditions simplify down to one. Furthermore, decentralized electronic records imply a consensus between the different copies of the record, so Bitcoin is a distributed consensus system.
What is the purpose of all this discussion? It may be useful legally – if the government tries to ban Bitcoin, we can argue that it just numbers in a database, and we don't believe that it's our fault that some people are crazy enough to accept them in exchange for dollars, weed and alpaca socks. It has pure philosophical value as well. But a deeper understanding of what Bitcoin is is most necessary if we want to take the idea. Rather than copying ideas many thousands of years old like currency, create something new entirely. In a future where most of the goods produced are non-rivalrous, like software and books and movies, it might make sense to have some non-rivalrous equivalent of a currency to reward the producers of such things without limiting how much they can be consumed. It's hard to tell exactly what such a thing would look like, but that's up to future inventors to decide. We might have a database of identities and reputations on a blockchain sometime in the future. We might even be using blockchains to enforce the constitutions of democratic institutions, whether governments, co-operatives or something in between. We already have Namecoin; if we take a step back and see what we have already created, the yet unseen possibilities are potentially endless.