Bitcoin has value because it is accepted as payment by many. The initial market value was achieved when people speculated, that because of its properties, the currency would be accepted by others later on.

When we say that a currency is backed up by gold, we mean that there's a promise in place that you can exchange the currency for gold. In a sense, you could say that Bitcoin is "backed up" by the price tags of merchants and currency exchangers – a price tag is a promise to exchange goods for a specified amount of currency. Of course, price tags may or may not be as long-term promises as those made by central banks or governments.

It's a common misconception that Bitcoins gain their value from the cost of electricity required to generate them. Cost doesn't equal value – hiring 1,000 men to shovel a big hole in the ground may be costly, but not valuable. Also, even though scarcity is a critical requirement for a currency to be useful, it alone doesn't make anything valuable. Your fingerprints are scarce, but that doesn't mean they have any exchange value.