Bitcoin is not backed by anything other than people believing that it has value. You could make that same argument about government-issued fiat money, but enough people believe in (for example) the US Dollar that its value is pretty well established.
One of the problems with bitcoin is that it's far more volatile in value than government-issued currency.
The best explanation of bitcoin mining I've seen is in the first video link posted by ireckon above. In short: bitcoin mining uses progressively more complex and computationally expensive cryptographic calculations to create new bitcoins and process transactions.
If you prefer a more thorough textual explanation:
New bitcoins are generated by a competitive and decentralized process called "mining". This process involves that individuals are rewarded by the network for their services. Bitcoin miners are processing transactions and securing the network using specialized hardware and are collecting new bitcoins in exchange.
The Bitcoin protocol is designed in such a way that new bitcoins are created at a fixed rate. This makes Bitcoin mining a very competitive business. When more miners join the network, it becomes increasingly difficult to make a profit and miners must seek efficiency to cut their operating costs. No central authority or developer has any power to control or manipulate the system to increase their profits. Every Bitcoin node in the world will reject anything that does not comply with the rules it expects the system to follow.
Bitcoins are created at a decreasing and predictable rate. The number of new bitcoins created each year is automatically halved over time until bitcoin issuance halts completely with a total of 21 million bitcoins in existence. At this point, Bitcoin miners will probably be supported exclusively by numerous small transaction fees.
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