firefox.exe is the name of the process that runs Firefox. It's normal that this process be present if Firefox is running. Having that process when Firefox isn't running is indicative of a problem.

If you get the error Firefox is already running but is not responding when starting Firefox, read this article.

A firefox.exe process with Firefox not running can be caused by a crash in Firefox. In this case, you should be able to end the process and it shouldn't come back until you start Firefox again. See Firefox crashes for tips on preventing crashes.

A firefox.exe process may also mean you have a virus called "Poison Ivy". Some symptoms of this virus include:

• When you end the process, it comes right back.
• The process is running when you start Windows.
• Your security software warns you about something happening on port 3460.

You should ensure that your anti-virus software is up to date and run a full system scan.

Based on information from Firefox.exe always open (mozillaZine KB)