Please explain the proof-of-work concept.
And how does it relate to Bitcoin mining in general, and to the proofs of work (aka shares) of mining pools?
Bitcoin miners find a random number (called a "Nonce") that when inserted into the current block makes the hash be below the current target. They then send that current block around the network and everyone checks their work (the proof-of-work) by hashing the block and checking if the result is below the current target. In mining pools, miners do the same thing, but the current target is much higher (so it's easier). When they find one of these, it's called "mining a share". A small percentage of the "shares" will actually have a hash below the actual target, and the pool will make it's 25BTC profit, then distribute it out to the miners who found a share (normally using some über-complicated method to prevent cheating/pool hopping)
Due to the very low probability of successful generation, this makes it unpredictable which worker computer in the network will be able to generate the next block.
This is what makes proof-of-work based cryptocurrencies like Bitcoin different from any other form of electronic money. With access to mining being open to all, there is no way to control (or corrupt) the transaction verification process.
For instance, let's say there was a law passed everywhere mandating that no payments to Wikileaks would be allowed -- that any miner including a payment to the Wikileaks payment address would be punished.
That is a form of centralized control and some miners would comply. But because other miners would continue verifying those transactions regardless, the worst that would happen is that payments to Wikileaks might get ignored by a block or three of miners who did comply with the law until another miner ignoring the law gets a block. The economic advantage goes to the uncorrupted miners (who recognize the laws of math and ignore the laws made by man).