How does proof-of-stake "mining" work? I know and understand the concept of proof-of-stake. Instead of requiring users to do a certain amount of power-intensive hashing "work", it requires you to own a certain stake of the currency in order for you to mine new coins. 

But how does the actual mining process work? In PoW mining, people "find" blocks by generating a hash that fulfills a certain condition. How can one mine a PoS block? Is a miner required? Or does just anyone holding coins get their share after a certain amount of time? 

In Peercoin specifically, is PoW and PoS mining done independently or do they interfere or influence one another?
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.