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A very interesting academic paper has been recently published by two researches from the Cornell University (abstract, full PDF): The Bitcoin cryptocurrency records its transactions in a public log called the blockchain. Its security rests critically on the distributed protocol that maintains the blockchain, run by participants called miners.
As others have said, due to the fear of you asking to be charged back, almost no seller is going to accept a credit/debit card in exchange for Bitcoins. If your bank allows you to get cash as a kind of "credit card transaction" (some do!) you might want to try that option and then pay in cash - you can always find people gladly selling in exchange for cash, and you can still pay monthly (in fiat) to your bank:
The algorithm of choice is two rounds of SHA256, which has completely arbitrary outputs and a massively unsearchable output space. To this end, there is no data mining or valuable calculation actually being done. Here's what's really happening: Every so often, the rules of the Bitcoin protocol set a difficulty number. This number sets a sort of target for a value miners are looking for.
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