To delete a category, first make sure that it has no dependent child categories and is not being used by a category field on a Content Type (See "Deleting Categories with Dependencies" below). Then select the checkbox(es) to the left of the obsolete categories and click on the delete button.
Deleting Categories with Dependencies
A category can not be deleted until all child categories under that category have been deleted first. A parent category can not be removed if it still is being used by a category field on a Content Type of any kind. First remove the category field(s) on all Content Type where it is used, then delete the category as stated above.
The example below shows the error messaging used when attempting to delete a category that is still the parent of a dependent child category, or is still being used by a category field on a Content Type.