By your description it appears you have a couple of separators in your bookmark list. Try deleting the separators to see if it helps. To do that open the bookmarks sidebar (in the view menu select Sidebar and then Bookmarks). In the bookmarks sidebar right-click on the separator and select Delete. Once both separators have been deleted do you still have this problem?

In addition to the above, Firefox sorts bookmarks between separators.
A possible use of it is to sort folders separately without using an extension.

See also http://kb.mozillazine.org/Sorting_and_rearranging_bookmarks_-_Firefox

If you are rearranging your bookmarks in the Organize Bookmarks / Library window, that is a temporary view, as soon as you close the Library window the new ordering is lost. You should use the Bookmarks Sidebar for rearranging bookmarks, as shown here. https://support.mozilla.com/en-US/kb/Sorting+bookmarks

It it only temporarily in the Library if you use the Views menu (what is in a name!) and not if you sort a folder via the right-click context menu of that folder. The later actually sort bookmarks and the former is meant for viewing purposes without changing anything. You can compare that with Windows Explorer or what happens if you sort items by clicking the header.

Removing the separators did the trick. Thanks to all, I didn't even know you could delete those bad boys. Thanks again.

Although I went through and removed all of the separators (there were several), I still have the problem. There are many bookmark folders and individual bookmarks in the list that are not in alphabetical order. What more can I do to organize them by name?

You can look at this extension:

• SortPlaces: https://addons.mozilla.org/firefox/addon/sortplaces/