There's two different versions of Flash, ActiveX for IE and the Plugin version for other browsers.

1.Download the Flash setup file from here:
Adobe Flash - 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.

• 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

I've followed these instructions explicitly with the same result. It does not work. Any more suggestions?

Jean

Do you see the Flash plugin (NPSWF32.DLL) in the C:\Windows\System32\Macromed\Flash\ folder?

No, it's in the sysWOW64/Macromed/Flash folder. Can I move or copy it to System 32?

Make sure that the Flash Player plugin is enabled. Go to Tools|Add-ons. Then click the Plugins tab. Scroll down to the Flash Player plugin and click on the Enable button.

HTH Ken

No, it's in the sysWOW64/Macromed/Flash folder. Can I move or copy it to System 32?

No! That's the same place I see the Flash 10.1 plugin NPSWF32.dll on my 64-bit Windows 7 system so leave it there!

Make sure that the Shockwave Flash plugin is enabled in Firefox "Tools -> Add-ons Plugins", like Ken said. I don't see it in your list of plugins.

P.S. For other possible solutions, see :
http://kb2.adobe.com/cps/191/tn_19166.html
Troubleshoot Flash Player installation | Windows