Group inboxes are exactly how they sound, PM inboxes for messages sent to groups. Unlike discussions in categories, whose permissions are determined at the category level, PM conversations can have more granular permissions.
If you check the Interaction settings on a group it can be given a custom inbound email address. Emailing that address will create a new PM, with the group and topic creator as participants.
This won’t appear in a normal category view though, you will receive notifications as for any PM and need to visit the group inbox to see them.