ArsTechnica - Bitcoin: inside the encrypted, peer-to-peer digital currencyThis article by Thomas Lowenthal on ArsTechnica provides a good overview on mining:
The usual solution to the double-spending problem is a trusted intermediary. PayPal makes sure that you can’t spend the same dollars twice by deducting them from your account before they get added to someone else’s account.
This centralized approach is the one that enigmatic creator Satoshi Nakamoto specifically tried to avoid in the original Bitcoin design. The idea was to use cryptography to create verifiable transaction records without the need to trust anyone but your own calculations.
The Bitcoin solution uses cryptography and an open transaction register.
By finding the newest solution to the proof-of-work problem, a Bitcoin client confirms the history of previous transactions and moved the transaction register forward, allowing new debits and credits to form part of the next block that can be mined to earn more coins. Future coins can’t be mined in advance, because the computation to find the new block (and hence create new Bitcoins) relies on the the chain of previous blocks and the history of transactions since the most recent block.