Bitcoin is a peer-to-peer digital currency. Peer-to-peer (P2P) means that there is no central authority to issue new money or keep track of transactions. Instead, these tasks are managed collectively by the nodes of the network.

For a detailed introduction to Bitcoin I'd like to refer you to their website, a recent omega-tau podcast (also featured on Thilo's Tech Radio, focussing more on the technological and mathematical aspects), or a recent EconTalk podcast (focussing more on the economic aspect), let me just point out a few properties of Bitcoin that make this so exciting for me:

• Hard as gold: While Bitcoin is obviously a virtual currency, it is in a way much more real than "regular" money, in that every Bitcoin is a scarce mathematical object, of which only a limited number exists, and which cannot be faked. In that regard, Bitcoin is a lot like gold. In fact, the process of discovering new Bitcoins (which happens at a controlled and decreasing rate of currently 50 coins every ten minutes) is even called "mining".
• Works like cash: Your Bitcoin wallet is a file on your computer containing private keys necessary for the cryptographic calculations necessary to spend your Bitcoins. Whoever holds that file, controls the wallet (so you'd better encrypt it). Unlike cash, however, it cannot be found again once lost: If you lose your wallet file, your Bitcoins are essentially destroyed (so you'd better back it up).
• No central bank: Because Bitcoins are like gold, it is not possible for anyone to create new money out of thin air in order to try to manipulate economic developments. You do not have to trust Angela Merkel or Ben Bernanke to make the right decisions in this area. It would be possible to change the rules of Bitcoin, but a majority of its users would have to agree to use the updated software, otherwise it just doesn't happen.
• No local banks: You do not have to go to any kind of service to open an account for you wherein to store your money in order to be able to participate in the Bitcoin economy. Everyone can run their own bank, being responsible (and solely responsible) for his own Bitcoins. All you need to do is download a piece of software. (This currently works very well, but it requires everyone to receive and maintain a copy of the whole transaction history of the whole network, which will hit a scalability wall once the system becomes more popular. At one point, there will be intermediaries necessary, but the general idea stays the same).
• Anonymity: A Bitcoin account is autonomously created by their owner. It is just a long number, not associated publicly with any particular person or organization. You have to disclose the number to the person you want to receive Bitcoins from, so depending on the type of transaction, they may be able to associate that number to you. But you never have to use the same receiving account twice, so this information is not too useful for tracking purposes (you do have to use the account again when you want to withdraw the Bitcoins it contains).
• Transparency: While you can choose to obfuscate the relationship between yourself and your accounts, you can also go the opposite way and publicly announce the account number. This allows two things to happen: Anyone can send you Bitcoins without having to contact you first (thereby protecting their own anonymity), and anyone can see exactly how many Bitcoins you are receiving and spending from that account. Both sounds like a good match for charitable organizations or political parties.
• Accessible to research: All Bitcoin transactions happen publicly for everyone on the network to see (this is necessary for everyone to be able to verify how much coins are in every account). This means that you have an accurate record of the coin flows in the system, much more accurate than you'd have with traditional currencies.
• Very cheap transactions: Sending Bitcoins around is free right now. This works because the massive computational efforts required to keep it running are compensated by "mining": Every ten minutes someone will underwrite all transactions that he saw in the last ten minutes, thus making it official, and in return will receive 50 Bitcoins that are newly minted in the process. This will eventually go away and be replaced by transaction fees, but those will be minuscule compared to what we have in mainstream e-commerce now. The reduced transaction costs enable all kinds of new micropayment services that would otherwise not be feasible.
• Last, but not least, it is very clever, and very cool: As someone interested in cryptography and distributed systems, I have to appreciate the beauty of the Bitcoin system. The concept of using digital signatures to state your intent to transfer money from your account to somewhere else has been around for some time, but the way Bitcoin uses the idea of a Proof of Work to solve the problem of preventing you from going around and spreading the same signature many times (thus double-spending) without a central clearing house is ingenious. What it basically does is make sure that somewhere on the network someone gets to decide which of your transactions are "official", and that someone is randomly selected, so that you cannot fake this seal of approval (short of separating your victim from the network for an extended period of time, or by controlling a majority of computing power in the whole system, all without the victim being alarmed by that).