Bitcoin is based around a public ledger called the block-chain. The ledger says who owns what; you can think of it as a list of account numbers and balances. You move coins between accounts by publishing a transaction with the source and destination accounts and how much to move. Every 10 minutes or so, the miners collect all the new transactions, validate them, and add them in a new block to the end of the public ledger. The clever part is that Bitcoin is distributed; there's no central entity that manages the ledger, and anyone can post updates to it. The Bitcoin protocol keeps it all sane.

"Mining" means three things. First, it means validating the transactions. That means they check the source and destinations exist, that the source account has the amount specified, and that the transaction has the proper authority to move the money. Authorities are expressed using strong public key cryptology. Because of this step, you can't spend coins that don't belong to you.

Second, the miner has to solve a hard puzzle called the "proof of work". Doing this is computationally expensive in a way that can't be faked. This step is used to automatically resolve disputes when two different miners try to post different but equally valid updates. In effect they are voting, and the proof of work makes votes expensive so no-one can stuff the ballot box.

Third, the miner gets rewarded for doing the previous two things with transactions fees, and some new coins that are created out of nothing. This is the only way new coins are created. The process is called "mining" because it's doing work to produce value, much as digging in a gold mine is.

The upshot is that bitcoins are valuable for much the same reasons gold is valuable: they both exist in limited supply, are hard to fake, and facilitate trade. They are both imperishable and finely divisible. Bitcoins have the advantage over gold that they weigh nothing and can be sent digitally around the world cheaply.