Drupal and permissions - avoiding 'The directory sites/default/files is not writable' error

Have you moved a Drupal site to another server and received the error 'The directory sites/default/files is not writable' ?  The site worked on the previous server, and the permissions seem the same. Ah, an easy fix...make the permissions for the files directory 777. Noooo!

Let's take a look at what the numbers in the permissions setting mean.