Where does the value of Bitcoin stem from? What backs up Bitcoin?
Bitcoins have value because they are useful and because they are scarce. As they are accepted by more merchants, their value will stabilize. See the list of Bitcoin-accepting sites.
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. Bitcoins, like dollars and euros, are not backed up by anything except the variety of merchants that accept them.
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 useful currency, it alone doesn't make anything valuable. For example, your fingerprints are scarce, but that doesn't mean they have any exchange value.