Here you can specify which categories users in this group can see, and for which categories they can manage the sub-categories.
Can View Categories
The 'Can View' checkbox for a particular category controls whether or not users in the group can see the category when browsing categories (for example under the Browse option or on the homepage). Unlike folder permissions, this does not affect which assets they can see, just which categories show up. By default, all categories are visible for all groups.
Note that when a parent category is unselected, then all its descendant subcategories are necessarily not visible, since the user will not be able to browse down to them.
Category visibility is hierarchical in the same way as attribute visibility (i.e. a category can only be hidden for a group if it is also hidden for public and logged in users).
Can Manage Subcategories
For a parent category, the 'Can Manage Subcategories' checkbox is used to give users permission to add, edit and reorder all the sub-categories of the selected category (but not the ability to edit or delete the category itself). Note that this permission is inherited for all the subcategories of the selected category, so the user will be able to manage all subcategories that are descendants of the parent, at any depth (so ticking the checkbox for a category will automatically tick the checkboxes for all descendant categories).
Permission to manage subcategories is also hierarchical in the same way as attribute visibility (i.e. a category can only be hidden for a group if it is also hidden for public and logged in users).
Once a user group has permission to edit a category, users in this group will see an Admin section in the left hand menu containing a link to the Categories admin page: