Navigate to People | Permissions (admin/people/permissions).

This page lists out all the permissions organized by a module and provides a column for each role. You can see that it's possible to allocate permissions in a very granular way—a key feature of Drupal:

Navigate your way through the permissions grid and locate the section containing all the permissions related to the Contact module:

If you want both visitors and existing account holders to be able to use the contact form, then you need to allocate the Use the site-wide contact form permission to the Anonymous user (marked as (1) in the preceding screenshot) and the Authenticated user (2).

Note that the Administrator role is automatically granted all available permissions whenever a module is enabled that defines permissions, so the check box is automatically ticked (3).

Next, move down to the Node module section.

Drupal is frugal with its allocation of permissions. For example, while the Node module allows both the Anonymous...