You need the 32 bit version of Flash Plugin for Firefox.

There are two different versions of Flash, the ActiveX version for IE and the Plugin version used by most other browsers.

1.Download the Flash Plugin version setup file from here:
Adobe Flash - 32-bit Plugin version.
Save it to your Desktop.

2. Close Firefox using File > Exit
then check the Task Manager > Processes tab to make sure firefox.exe is closed,
{XP: Ctrl+Alt+Del, Vista: Shift+Ctrl+ESC = Processes tab}

3. Then run the Flash setup file from your Desktop.

4. Start Firefox and test your installation here: https://www.adobe.com/software/flash/about/

• On Vista and Windows 7 you may need to run the plugin installer as Administrator by starting the installer via the right-click context menu if you do not get an UAC prompt to ask for permission to continue (i.e nothing seems to happen).
  

See this:
http://vistasupport.mvps.org/run_as_administrator.htm

edited to fix the download link for Flash

The link was dead but installing the 32bit version is what worked for me. Thanks, I was about ready to go postal.

The 32 bit didn't work for me either. None of the solutions on Firefox help have resolved this issue. I am on FF 8.0 and trying to get FP 11 to work. I've tried both 32 and 64. I am giving up on FF.

Try this download link.
http://download.macromedia.com/pub/flashplayer/current/install_flash_player_32bit.exe