What is proof-of-work? 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)