Having a common currency such as the Euro is a great way to bring down geographical barriers and create a shared economy. And with audacity that feels like it's drawn straight out of one of William Gibson's cyberpunk novels, that's what Bitcoin (free) sets out to do.
Bitcoin works like a peer-to-peer file-sharing application, only instead of transferring files, the nodes on the network transfer data about financial transactions. Each node on the Bitcoin network has its own unique identifier. To send someone money, simple use the client to specify an amount, paste in their ID, and the money is sent on its way.
Of course, before you can do that, you need to get some Bitcoins to send. This isn't monopoly money: There are only so many Bitcoins in existence, and they have value that can be measured in more common currency (US Dollars, for example). The Bitcoin economy is still forming, so Bitcoins are being slowly created, or "mined" in Bitcoin parlance. Since one of Bitcoin's key tenets is that there should be no central issuing authority, the clients are doing the mining, effectively creating new money out of thin air.
At first glance, that seems very much off - kind of like printing your own Dollar bills at home. But mining new Bitcoins is a temporary affair - the rate of mining is hard-coded into the application, and will slowly taper off until at last, no more coins will be mined and the system will stabilize at 21 million Bitcoins.
Mining your own coins is an extremely slow process. It's not supposed to be fast. To actually start using Bitcoin as a form of online currency, you should first find out if there are any businesses that you'd like to work with which accept Bitcoin payments. Bitcoin's community wiki lists dozens of businesses that accept this new form of currency, with more being added constantly.
Once you've found someone you'd like to pay, you should purchase some Bitcoins to pay them with. This is done using an online currency exchange. There are a number of Bitcoin currency exchanges, again, all listed on Bitcoin's Wiki. At first, paying with "real" money for a virtual form of currency may leave you with an uneasy feeling. But once you've used your Bitcoins to pay for a valuable service, you would see that Bitcoin is just like any other form of currency, only it's not centrally controlled in any way.
One last caveat: Bitcoins are stored in a "wallet," which is nothing but an encrypted file on your computer. There is no central bank, and losing this file means you've lost all of your Bitcoins. So if you do use Bitcoin for anything serious, be sure to constantly back up this file to protect. Encryption might not go amiss either, as the Allinvain heist suggests. Digital wallets are vulnerable to hacks, and Symantec recently reported a Bitcoin-stealing Trojan.