In Nextcoin, proof of stake is used. So the "mining" process there is just about holding coins and leaving your computer on. It doesn't involve powerful CPUs.
Each block (every 60 seconds), a random Nextcoin is selected to be the next "miner". There are 1 billion coins so the odds of a single wallet being selected is the number of Nxt in that wallet divided by 1 billion. (Also, it's possible to calculate and agree on who that node is so the transactions need only be sent to that particular wallet.)
If a node with the selected wallet is running, it will collect the transactions, make a block, and send it to the rest of the network and collect the fees. If the computer is turned off, however, then the entire network will have to select a different nextcoin to make the transaction. This time, the unresponsive wallet will be ignored. The network would suffer in that the time to make a block is decreased, but the thought is that people wouldn't leave their computers off if they have a lot of NXT because they're missing out on all the fees that they could have collected.
If you only have a few NXT, you can leave your computer off: You probably wouldn't have collected much fees anyway. But, your odds of being selected were low so it probably won't decrease transaction times much.