Something probably prevented it from shutting down properly. First, you need to end the running Firefox process, then figure out what is preventing Firefox from shutting down.

Try the instructions at https://support.mozilla.org/en-US/kb/Firefox-hangs#w_firefox-hangs-when-you-quit-it

A possible cause is security software (firewall) that blocks or restricts Firefox or the plugin-container process without informing you, possibly after detecting changes (update) to the Firefox program.

Remove all rules for Firefox and the plugin-container from the permissions list in the firewall and let your firewall ask again for permission to get full unrestricted access to internet for Firefox and the plugin-container process and the updater process.

See:

• https://support.mozilla.org/kb/Server+not+found
• https://support.mozilla.org/kb/Firewalls

Chris_Ilias: Maybe I wasn't clear enough. There are NO Firefox processes running. I start Firefox. The process starts but there is NO window. It's not the case that it continues running after I attempt to shut it down. There is never any window displayed to attempt to shut it down. I have to kill the process with Task Manager or System Internal's (now Microsoft's) Process Explorer.

cor-el: I turned off the Firewall completely and the problem persists.

How about the instructions in the Firefox won't start - find solutions article then?

Chris_Ilias: Nope. Not much there that applies to my situation. And the most disappointing thing is that Firefox is totally silent. No log file, nothing in the event log.

Update: When Windows XP is started in Safe Mode with Networking, Firefox displays its window and operates correctly. After shutting down Windows and restarting it in normal mode (i.e. not Safe Mode), Firefox displays its window. Very strange that Firefox first had to run while Windows was in Safe Mode before it would behave correctly when Windows is not running in Safe Mode.

I'll keep an eye on this and reboot a number of times to see if Firefox keeps working correctly.

Update: Apparently, the contents of files and folders in the Windows XP folder \Documents and Settings\xxxx\Application Data\Mozilla (where xxxx is the user) was corrupted. Logged in as the Administrator, I uninstalled Firefox (using Add/Remove Programs control panel applet) and then re-downloaded Firefox and re-installed. Mysteriously, this did not fix the problem while logged on as Administrator until I re-booted once in Windows Safe Mode. But once I did the Windows Safe Mode thing and restarted Windows in normal mode, Firefox worked when logged in as Administrator. It continued to fail when logged in as other users. When you uninstall Firefox and it asks you if you want all your stored data removed, answering yes apparently only removes that data for the user you are currently logged in as. It does not look for and remove the \Documents and Settings\xxxx\Application Data\Mozilla contents for all users. Once I manually removed these folders for all users (while logged in as the Administrator), Firefox worked normally for all users.