Removing/Recovering Data from a Hard Drive

A couple of days ago on 4chan /b/, a thread about CP (nothing new there) segued into a discussion about the nature of hard drives, and different methods of 'permanently' removing data from them.  I thought some of the things mentioned were pretty interesting, but some of the questions about hard drives went unanswered.  So, I bring you these questions, hoping there will be a more in-depth discussion here.

tl;dr -- How is 'deleted' data recovered from a disk, and what are some ways of 'completely' removing data from the disk?

The US Government recommends you write over the data 7 times.