Add an Administrator? I've added a new Drupal user, so how do I make them an administrator of the site?

I can't seem to find a setting to do this? User roles are Authenticated and Anonymous.

But the default login username definitely has some kind of administrator permission. How do I grant this same privilege to a new user? 

In Config > People there is setting Administrator role which is a select menu, but it only has the disabled option :S
The 'super user' (user 1) has privileges that you won't be able to completely emulate for other users. There are hard-coded bypasses in functions like user_access() that allow that user to do pretty much anything they want.

Permissions are granular in Drupal, so the idea is you give a certain role a certain set of permissions based on what they should be allowed to do.

To get as near as possible to another role being able to do everything the super user can, you just need to go to admin/people/permissions and tick all of the permission boxes for the administrator role. I'd advise checking each permission (if only this once) as you do this, to make sure you're definitely giving that role the access you expect.

When you then assign the administrator role to other users, they'll essentially be able to do anything on the site that you've allowed them to do through the permissions page.