So the miners aren't really mining. They are doing work by completing the Bitcoin transactions. Rewarded with the newly issued Bitcoins. When a block of work is done they are paid a certain amount of coins.
Calling them workers instead of miners would clear up a lot of confusion. But then you would miss out on the gold fever mentality, mob rush.
I think you may need to start
here before jumping to the wiki.
And yes, what you described is mostly correct.
EDIT: Why da fuck doesn't bitcoin.org have SSL? It doesn't even redirect to plain HTTP, just gives a Server Not Found browser error page. Awesome!