In Coppermine, uploaded files go into albums. Albums can contain files, but not sub-albums. Albums can go into categories. Categories can contain sub-categories and albums, but they can not contain files.
Subsequently, categories are used to organize your gallery - use them to group your albums into logical units. It's up to you as gallery admin to provide an organizational structure of categories that provides your end users with a logical structure that makes it easy to navigate.
The category manager is available for the admin only. Only the admin can create categories.
This panel allows you to edit your categories.
button
allows you to edit the title, description and parent category of an
existing cetegory.
button
allows you to delete a category. Deleting a category does not delete the
albums and files it contains. These are moved to the "Root"
category if it is a subcategory that is deleted or to the root of the category list if it is a parent category that is deleted..
and 
buttons allow you to
order your categories."User galleries" is a special category. It is not visible unless you have some users that have created their own gallery. It can't be deleted but you can edit its title and description by using the button. Determine wether users are allowed to have personal galleries that show up within the "User galleries" using the groups control panel.
You can specify how you want categories sorted in coppermine: alphabetically (instead of a customized order) by setting "Sort categories alphabetically" to "Yes". This setting is available both in coppermine config and the category manager. If you enable alphabetical sorting, the move up and move down arrows that normally let you manually sort the categories will disappear. Disable this feature if you want to organize your categories in some other order.
You can only assign a picture to the category only if you have an album with images nested directly within it.
In coppermine versions before cpg1.5.x, only the admin was allowed to create albums within public (i.e. admin-created) categories. Starting with cpg1.5.x, this has changed: by default, non-admin users can't create albums, but the administrator can decide to allow them to create albums. This can be set by category and user group, so you can allow your users to create albums in category "A" but disallow them to create albums in category "B". This is why there is an additional list box in the category control panel named "Group(s) allowed to create albums in this category" where you (as admin) can specify the groups you wish to allow. The admin group is not being listed there, as the admin can create albums anyway and doesn't need specific permissions