Mining bitcoin is a bit of a misnomer. You don't actually mine them at all. Instead you are rewarded with them for solving mathematical problems.
Each time somebody makes a bitcoin transaction (I pay you some bitcoin), that transaction goes into a queue together with all other transactions that are waiting.
A bitcoin "miner" will grab a bunch of potential transactions from the queue and try and solve a maths problem that will allow her to bundle them up in a single "block" and add them to the list of transactions that are valid. If she is the first person to do this then she gets awarded with some new bitcoins (created out of nowhere) for doing so. She also gets rewarded with a "transaction fee" which the participants in the transaction will pay.
Her new block of transactions will then get added to the list of valid transactions "the chain" and transmitted to everybody that has a copy of the chain. As the chain is monitored by a large amount of people and each block depends on every block prior to it, it is nigh-on impossible to hack it or create a fraudulent transaction.
As more powerful hardware comes along, the maths puzzle gets harder and it gets harder to "mine" new bitcoin.
Eventually, and by design, there will be no new bitcoin available and the only reward for miners will be the transaction fees. However this will be set at a level that still makes it worthwhile to mine. Once this happens bitcoin will be valued for its rareity - just like Roman coins are.