History and cookies can be removed or not recorded several ways;

• 1) Running Firefox in Private Mode (nothing is recorded)
• 2) Privacy settings
  (Type about:preferences#privacy<enter> in the address bar.)
• 3) Many disk cleaners and anti-virus type programs can remove data from browsers. Open their settings and have them leave the browsers alone.
• 4) Mal-ware.

As I said, I have checked privacy setting and it is marked to remember history. Firefox is not being run in private mode, and no one else uses this computer. I have not run any disk cleaners, but I suppose Mal-ware could be an option though nothing is being picked up by my mal-ware program. Have you one you would recommend?

You may have ad / mal-ware. Further information can be found in the Troubleshoot Firefox issues caused by malware article.

Run most or all of the listed malware scanners. Each works differently. If one program misses something, another may pick it up.

Start Firefox in Safe Mode {web Link} by holding down the <Shift>
(Mac=Options) key, and then starting Firefox. Is the problem still there?

While still in Safe Mode,

Type about:support in the address bar and press Enter. Under the main banner, press the button; Copy Text To Clipboard.. Then in the reply box at the bottom of this page, do a right click in the box and select Paste. This will show us your system details. No Personal Information Is Collected.

You can check for problems with the places.sqlite database (bookmarks, history) in the Firefox profile folder.

• Places Maintenance: https://addons.mozilla.org/firefox/addon/places-maintenance/

See also:

• http://kb.mozillazine.org/Bookmarks_history_and_toolbar_buttons_not_working_-_Firefox
• https://support.mozilla.org/kb/Bookmarks+not+saved#w_fix-the-bookmarks-file