Bitcoins are a decentralized, peer-to-peer virtual currency. There are other digital currencies that offer an alternative to government fiat currencies, but the popular ones like e-gold tend to suffer eventual demise from gaining just as much momentum with money launderers and other nefarious characters as they do regular users. And they are all privately controlled, meaning they are subject to the control of a single entity, just like a government currency.
Being peer-to-peer and open-source, Bitcoins is protected from centralized control and centralized manipulation both at once. There is no single entity to shut down, there is no single entity to possibly manipulate the system, and the source code the system is built on is freely available for inspection. The currency is not backed by gold or any other physical resource in finite supply, but the supply of bitcoins themselves is limited – there will only be a maximum of 21 million bitcoins in existence, ever, by design. A limited supply means that this currency, like its gold-backed cousins, is protected from inflation (the effect of rising prices due to an increase in the currency supply).
The value of bitcoins depends on their acceptability as trade for goods and services, just like all currencies. A dollar is just a piece of paper; it has value because it is accepted as payment by someone that has something you wish to buy. Bitcoins is a new project still in beta, but the list of merchants accepting bitcoins is growing.
You can download the Bitcoins application for Windows, Mac or Linux. There is nothing to configure; once installed the program automatically generates a bitcoins address for your account. You can get bitcoins either by selling goods and services for them, or by contributing CPU resources from your machine to the project, which are used to generate the cryptographic hashes the system is built on. Turn on Settings -> Generate Coins, and when your system is idle it will use your unused processor cycles to calculate hashes for the project. It takes a very long time to generate bitcoins this way, but it costs you nothing because your system was idle anyway.
Each bitcoins address is a private/public key pair of cryptographic hashes. Bitcoins are associated throughout the network with the bitcoins address/hash that owns them, but without the private key that matches that address, you won't have access to spend the bitcoins and they would be lost forever. For anonymity, a new address is used for each transaction, and since the bitcoins are assigned to the address, not to you, it's safest to do regular backups to prevent losing bitcoins permanently if your hard drive were to crash. A free dropbox account is perfect for this, because it's encrypted and stored in the cloud. Although the bitcoins application doesn't currently have a way to change the location of your wallet (so it would all be backed up automatically by storing it directly in your dropbox account), it's trivial to make a cron job or scheduled task to copy your wallet over to dropbox on a periodic basis. (On Windows, your wallet.dat file is stored in your %appdata% folder.)
Since it can take weeks, months or even more than a year to generate a block of 50 bitcoins (depending on the power of your CPU or GPU), this is not a get-rich-quick scheme. Instead it is merely promoting an open-source virtual currency project that may or may not eventually become a widely-used alternative currency. You may never end up spending a single Bitcoin, or the whole thing may eventually be abandoned from a lack of popularity, but at least you're not really losing anything because it only uses idle system resources that would have gone to waste anyway.
Eventually, all the bitcoins will have been generated and there will be no new ones, making buying/selling/trading the only way to get them. With a finite supply, their value will increase over time relative to other currencies which have no limit of supply (like the dollar), resulting in a gradual deflation of prices. Meaning that although there will never be new bitcoins once all of them have been generated, something that costs 5 bitcoins today may only cost 4 bitcoins next week.
Right now, this project is gaining a lot of popularity especially in the technology sector. It's still a little early and a little geeky to be easily understandable by the layperson, but it's also by getting in early that you can have a better chance of acquiring bitcoins whether by generating them or by trading goods and services for them. Like gold, bitcoins which have a limited supply will only increase in value over time. Of course, since unlike gold they have no intrinsic value, the whole thing may flop and become yesterday's fad. Who knows?
But if, like me, you've got a spare computer that is rarely used anyway, it doesn't hurt to let it do a little work for you instead of sitting idle. Or you can run it on a computer that you do use, since the Bitcoins application configures itself as low priority and will only use idle system resources. Either way, why let all that processing power go to waste? If you'd prefer a less self-serving way to use idle CPU cycles, there's always the SETI project.