Are you installing the correct version of Adobe Flash?

There's two different versions of Flash for Windows PC's. The ActiveX version for Internet Explorer and the Plugin version for almost all other web browsers, you need the Plugin version for Firefox.

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 followed your instructions, thanks for posting. It appeared the problem was still there when I selected a news article. In a place I believe a video may have been posted, there was a message 'plugins disabled, click here to fix'. I clicked to fix. The only plugin relating to flash was one called 'shockwave flash'. I enabled that and the Java plugins, everything except 'Hulu desktop?'. Videos now play, but as was happening lately, they are paused quite often. While there I selected the 'No Script' add-on.

I will try to install the adobe flash once again.

Sorry, I'm not sure what you have going on there. Your More system details... to the right of your initial posting showed no extensions at all as being available / enabled for Firefox. Evidently you had disabled all your Plugins; NoScript will block them from being active for a web page until you enable them, but it won't completely disable Plugins to the point of this forum's scan not seeing your Plugins when you "Ask a new question".

You need to learn how to use the NoScript extension. IMO, NoScript is almost more complicated to learn to use than the entire Firefox program. It is an excellent addition to Firefox's already good "security", but it can be a royal pain until you get all the "permissions" saved for the websites you visit routinely.
http://noscript.net/faq
http://noscript.net/screenshots

Hi edmeister, I have the plugin for IE. Do you mean I can also install the Firefox version? Will they not crash? Or do I have to opt for one and unistall the other? Thanks.

I had the same problem and I think I solved it. The only thing that is different in save mode is that extensions and moduls aren't allowed. So why not to allow them? Go to tools then to add-ons and there you should find a list of extensions and moduls. Allow them all. I did it and now everything works normal. Try that...(and sorry for my English)

Thank you Dalda. Your English is excellent.