There are two different versions of Flash for Windows, the ActiveX version for IE and the Plugin version used by Firefox and 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

It worked as stated. Why didn't the Firefox plug-in work???

You probably installed a 64 bit version of the Flash plugin for Firefox.
Firefox is a 332 bit application, so you need 32 bit versions of a plugin to make it work.

You can find the 32 bit version of the Flash Player installer for a 32 bit Firefox version here (scroll down to Flash Player (32 bit)):

• http://www.adobe.com/special/products/flashplayer/fp_distribution3.html