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.
13How can a network with different incentives agree on a random number in a way that can't be manipulated? – ike – 2014-12-22T22:44:45.617
3also, couldn't i build a chain where i'm the only miner because i was the only person online at those moments? wouldn't my chain win against the current chain? – Luca Matteis – 2015-07-24T11:29:03.357
so as I understand, a node with lesser stake coin will have less probability to be a miner right ? – hqt – 2018-03-17T15:15:06.890
1The best explanation I found [until now] – Aminadav Glickshtein – 2018-03-19T07:36:47.290