How do I enable and use the password module for Drupal 6? I've uploaded the module to my site, but it doesn't appear in the list of modules so I can't enable it. Does this module work differently than other modules?

https://drupal.org/project/password
No, this module works as any other module. See my screenshot as I can see it in my module listing - 

Are you sure you have it in sites/all/modules/ ? Are the permissions correctly set?