Seiler gave you the correct information. Even though those categories are “blogroll” categories they are both edited and deleted from within the Manage -> Categories screen. Note that you can also add them from this screen as well as add them from the “add link” dialog (simply as a convenience).
To use that right-side “categories widget” in the “Blogroll -> Add link” screen, make sure that you have “expanded” the widget (click the " " sign if it is present - if the “-” sign is showing it is already “expanded”). Then, to add a new blogroll category, just type in the box (and you can add more then one at once, by separating them with a comma).
If you somehow got some in here you don’t want, you can then delete those categories from the “Manage -> Categories” screen - just click the “delete” link on the right side of the screen for the category in question.
Read the notes on the screen so you know what happens if you delete a category that has a link assigned to only that category (hint: the link will then be set to “link”). Note that posts that are assigned to a category you have deleted will be “automagically” set to the “uncatagorized” category.
Have you read the Codex.wordpress.com documentation on Deleting a Category (and the rest of the page on managing links? While that documentation was written for an older version of WordPRess, it works the same (the “links” main menu item is now called “blogroll”, but the “Manage -> Categories” screen still works as described.)
The “currently showing” dropdown is only a way to filter which categories’ links are displayed - if there are no links *assigned to a category, you can’t “show” them, right? You cannot edit/delete a category (even a “blogroll” category) from that screen - the “delete” is only for the posts themselves. All category management happens, instead, as seiler said, and I’ve repeated in more detail above.
If this is still confusing you, why don’t you post a link to your blog and point out where the problem lies and what you are trying to do so we can understand better where you are having trouble?