Hello,
These files are all related to the registration and the file name itself pretty much sums it up
If you still don't get it, I will briefly explain them below:
approvals.php - This is when user registers and their account requires approval
auto.php - This is when the user signup process is automated (Automatically logged in)
login.php - Registration completed, you may now login with the details
moderator.auto.php - This is sent to the moderator if a new user signs up (User signed up and logged in automatically)
moderator.login.php - This is sent to the moderator if a new user signs up (User signed up and logged in automatically)
moderator.verify.php - This is sent to the moderator if a new user signs up (User signed up and requires moderation)
I am really sorry but unfortunately I cannot explain this in layman terms because in order for you to achieve this you need to add php codes / hacks on the theme file. I am not too sure how else to explain to you.