It's very hard to explain if you start from the code and descriptions of the Bitcoin system itself, as it solves problem people usually have never thought about, or even conflicts knowledge in the fields of computer science, cryptography and economics. It's as if you had to explain a computer science problem to people who haven't seen computers before. So the explanation of the system requires an explanation of the goals. The whitepaper doesn't go into this, but satoshi's first posts.
The Byzantine Generals' Problem
A number of Byzantine Generals each have a computer and want to attack the King's wi-fi by brute forcing the password, which they've learned is a certain number of characters in length. Once they stimulate the network to generate a packet, they must crack the password within a limited time to break in and erase the logs, lest they be discovered. They only have enough CPU power to crack it fast enough if a majority of them attack at the same time.
They don't particularly care when the attack will be, just that they agree. It has been decided that anyone who feels like it will announce an attack time, which we'll call the "plan", and whatever plan is heard first will be the official plan. The problem is that the network is not instantaneous, and if two generals announce different plans at close to the same time, some may hear one first and others hear the other first.
They use a proof-of-work chain to solve the problem. Once each general receives whatever plan he hears first, he sets his computer to solve a difficult hash-based proof-of-work problem that includes the plan in its hash. The proof-of-work is difficult enough that with all of them working at once, it's expected to take 10 minutes before one of them finds a solution and broadcasts it to the network. Once received, everyone adjusts the hash in their proof-of-work computation to include the first solution, so that when they find the next proof-of-work, it chains after the first one. If anyone was working on a different plan, they switch to this one, because its proof-of-work chain is now longer.
After about two hours, the plan should be hashed by a chain of 12 proofs-of-work. Every general, just by verifying the difficulty of the proof-of-work chain, can estimate how much parallel CPU power per hour was expended on it and see that it must have required the majority of the computers to produce in the allotted time. At the least, most of them had to have seen the plan, since the proof-of-work is proof that they worked on it. If the CPU power exhibited by the proof-of-work is sufficient to crack the password, they can safely attack at the agreed time.
The root problem with conventional currency is all the trust that's required to make it work. The central bank must be trusted not to debase the currency, but the history of fiat currencies is full of breaches of that trust. Banks must be trusted to hold our money and transfer it electronically, but they lend it out in waves of credit bubbles with barely a fraction in reserve. We have to trust them with our privacy, trust them not to let identity thieves drain our accounts. Their massive overhead costs make micropayments impossible.
A generation ago, multi-user time-sharing computer systems had a similar problem. Before strong encryption, users had to rely on password protection to secure their files, placing trust in the system administrator to keep their information private. Privacy could always be overridden by the admin based on his judgment call weighing the principle of privacy against other concerns, or at the behest of his superiors. Then strong encryption became available to the masses, and trust was no longer required. Data could be secured in a way that was physically impossible for others to access, no matter for what reason, no matter how good the excuse, no matter what.
It's time we had the same thing for money. With e-currency based on cryptographic proof, without the need to trust a third party middleman, money can be secure and transactions effortless.
One of the fundamental building blocks for such a system is digital signatures. A digital coin contains the public key of its owner. To transfer it, the owner signs the coin together with the public key of the next owner. Anyone can check the signatures to verify the chain of ownership. It works well to secure ownership, but leaves one big problem unsolved: double-spending. Any owner could try to re-spend an already spent coin by signing it again to another owner. The usual solution is for a trusted company with a central database to check for double-spending, but that just gets back to the trust model. In its central position, the company can override the users, and the fees needed to support the company make micropayments impractical.
Bitcoin's solution is to use a peer-to-peer network to check for double-spending. In a nutshell, the network works like a distributed timestamp server, stamping the first transaction to spend a coin. It takes advantage of the nature of information being easy to spread but hard to stifle. For details on how it works, see the design paper at
http://www.bitcoin.org/bitcoin.pdfThe result is a distributed system with no single point of failure. Users hold the crypto keys to their own money and transact directly with each other, with the help of the P2P network to check for double-spending.