I've been running a bitcoin only e-commerce business for 6 months now.

Previous currencies might have involved cryptography, but not to the extent that Bitcoin employs it.

What is special about it, is that the cryptographic schemes behind Bitcoin allows it to be DECENTRALIZED. There is no company, or body that can be swayed into changing the allocation of the currency. The bitcoin protocol is truly impressive in the way that it is almost it's own autonomous institution, providing value to humanity, and thus being supported by human activity.

Here's a brief run down of how it works, with some shortcuts.

1. "Bitcoin" is a decentralized means of tracking and assigning wealth or economic value. Bitcoin is a software protocol, computer network, idea, community, movement, etc.

2. A "bitcoin" is a unit of the currency that is moved around by the Bitcoin network.

3. Central to Bitcoin is a public ledger, known as the Block Chain. Roughly every 10 minutes, a new "block" is added to this chain or ledger. This ledger records all of the transaction that have taken place in the last 10 minutes, and what quantities of bitcoin currency are now held at different public addresses.

4. A public address is a a 27 to 34 character string of uppercase, lowercase letters and the digits from 0 to 9. AKA base 58.

5. Each public address has a corresponding private key. Whoever has this key, may spend the coins that are held at this private address. Private keys are 51 characters long in the same format as a public address.

6. To spend an amount of bitcoin, you must use your private key to cryptographically sign the transaction, sending your bitcoin to another address.

7. This message or transaction is then broadcast to the network, and the computers in the network begin working to write into the block chain (or public ledger) that your address no longer has the amount that was sent, but that that amount is now held at the receiving address.

8. Each new set of transaction is recorded on the block chain, every 10 minutes as mentioned above.

9. All of the computers that are working to write new blocks to the block chain, are known as miners.

10. These computers are all racing to solve a cryptographic puzzle, which is required to write the new block.

11. The computer that solves the puzzle, and writes the new block receives an award of newly created bitcoin.

12. The reward associated with each block began at 50 in 2009, is now 25, and will halve every 4 years, until an ultimate quantity of 21 million bitcoin are created.

13. This transparency is a large part of the value created by Bitcoin, in that the rate of creation, and current amount in existence is known.

That is my best attempt at translating what has taken me months to wrap my head around, and the implications go much further. I'm happy to answer other questions about Bitcoin; the system, the community, or business challenges and opportunities associated with it.