An address is a hash of a public key, which is derived from a private key. As long as you have the corresponding private key in your wallet, when you start bitcoin any payments sent to your addresses will show up normally (as soon as you download the blocks that contain the transactions).
I can confirm that Bitcoin actually does work like this in practise. My computer experienced a big meltdown and I could not run the Bitcoin client for a couple of days. During that time I received a payout from somebody. When I restarted bitcoin that payment showed up once it had enough time to catchup on the block-chain.
While my computer was down, I was able to confirm the payment had been sent by searching for my address on http://blockexplorer.com/