The proof-of-work chain is a solution to the Byzantine Generals' Problem. I'll try to rephrase it in that context.
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, otherwise they will be discovered and get in trouble. 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 all agree. It has been decided that anyone who feels like it will announce a time, and whatever time is heard first will be the official attack time. The problem is that the network is not instantaneous, and if two generals announce different attack times 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 attack time he hears first, he sets his computer to solve an extremely difficult proof-of-work problem that includes the attack time in its hash. The proof-of-work is so difficult, it's expected to take 10 minutes of them all working at once before one of them finds a solution. Once one of the generals finds a proof-of-work, he broadcasts it to the network, and everyone changes their current proof-of-work computation to include that proof-of-work in the hash they're working on. If anyone was working on a different attack time, they switch to this one, because its proof-of-work chain is now longer.
After two hours, one attack time 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 that much proof-of-work in the allotted time. They had to all have seen it because the proof-of-work is proof that they worked on it. If the CPU power exhibited by the proof-of-work chain is sufficient to crack the password, they can safely attack at the agreed time.
The proof-of-work chain is how all the synchronisation, distributed database and global view problems you've asked about are solved.
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For greater privacy, it's best to use bitcoin addresses only once.
We should have a gentleman's agreement to postpone the GPU arms race as long as we can for the good of the network. It's much easer to get new users up to speed if they don't have to worry about GPU drivers and compatibility. It's nice how anyone with just a CPU can compete fairly equally right now.
I'm happy if someone with artistic skill wants to contribute alternatives. The icon/logo was meant to be good as an icon at the 16x16 and 20x20 pixel sizes. I think it's the best program icon, but there's room for improvement at larger sizes for a graphic for use on websites. It'll be a lot simpler if authors could make their graphics public domain.
The proof-of-work is a Hashcash style SHA-256 collision finding. It's a memoryless process where you do millions of hashes a second, with a small chance of finding one each time. The 3 or 4 fastest nodes' dominance would only be proportional to their share of the total CPU power. Anyone's chance of finding a solution at any time is proportional to their CPU power.
When there are multiple double-spent versions of the same transaction, one and only one will become valid.
Difficulty just increased by 4 times, so now your cost is US$0.02/BTC.
I anticipate there will never be more than 100K nodes, probably less. It will reach an equilibrium where it's not worth it for more nodes to join in. The rest will be lightweight clients, which could be millions.
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.
The network is robust in its unstructured simplicity. Nodes work all at once with little coordination. They do not need to be identified, since messages are not routed to any particular place and only need to be delivered on a best effort basis. Nodes can leave and rejoin the network at will, accepting the proof-of-work chain as proof of what happened while they were gone. They vote with their CPU power, expressing their acceptance of valid blocks by working on extending them and rejecting invalid blocks by refusing to work on them. Any needed rules and incentives can be enforced with this consensus mechanism.
For future reference, here's my public key. It's the same one that's been there since the bitcoin.org site first went up in 2008. Grab it now in case you need it later. 
Bitcoin - Open source P2P money
Bitcoin is an innovative payment network and a new kind of money. Find all you need to know and get started with Bitcoin on bitcoin.org.
It is a global distributed database, with additions to the database by consent of the majority, based on a set of rules they follow:
- Whenever someone finds proof-of-work to generate a block, they get some new coins
- The proof-of-work difficulty is adjusted every two weeks to target an average of 6 blocks per hour (for the whole network)
- The coins given per block is cut in half every 4 years
The proof-of-work also solves the problem of determining representation in majority decision making. If the majority were based on one-IP-address-one-vote, it could be subverted by anyone able to allocate many IPs. Proof-of-work is essentially one-CPU-one-vote. The majority decision is represented by the longest chain, which has the greatest proof-of-work effort invested in it. If a majority of CPU power is controlled by honest nodes, the honest chain will grow the fastest and outpace any competing chains. To modify a past block, an attacker would have to redo the proof-of-work of the block and all blocks after it and then catch up with and surpass the work of the honest nodes. We will show later that the probability of a slower attacker catching up diminishes exponentially as subsequent blocks are added.
The average total coins generated across the network per day stays the same. Faster machines just get a larger share than slower machines. If everyone bought faster machines, they wouldn't get more coins than before.
It's not a problem if transactions have to wait one or a few extra cycles to get into a block.
There is no way for the software to automatically know if one chain is better than another except by the greatest proof-of-work. In the design it was necessary for it to switch to a longer chain no matter how far back it has to go.
The guy who received the double-spend that became invalid never thought he had it in the first place. His software would have shown the transaction go from "unconfirmed" to "invalid". If necessary, the UI can be made to hide transactions until they're sufficiently deep in the block chain.
As computers get faster and the total computing power applied to creating bitcoins increases, the difficulty increases proportionally to keep the total new production constant. Thus, it is known in advance how many new bitcoins will be created every year in the future.
Bitcoins have no dividend or potential future dividend, therefore not like a stock.
More like a collectible or commodity.
It can already be used for pay-to-send e-mail. The send dialog is resizeable and you can enter as long of a message as you like. It's sent directly when it connects. The recipient doubleclicks on the transaction to see the full message. If someone famous is getting more e-mail than they can read, but would still like to have a way for fans to contact them, they could set up Bitcoin and give out the IP address on their website. "Send X bitcoins to my priority hotline at this IP and I'll read the message personally."