Your add-ons list has multiple downloaders. I wonder whether they might be interfering with one another? Try disabling all but one at a time. You can do that here:

orange Firefox button (or Tools menu) > Add-ons > Plugins category
orange Firefox button (or Tools menu) > Add-ons > Extensions category

By the way, the Flash player was updated in the past 48 hours. Do you recall seeing the update? You can check your current version here: http://www.adobe.com/software/flash/about/. You might need to restart Firefox or Windows to get all your add-ons on the same page.

I just tried everything you suggested but it didn't change anything.It acts as if it is downloading but at an accelerated speed but when I try to play the download all I get is a black screen with no audio.

Here's another idea. Starting in Firefox 21, MPEG media can be played directly by Firefox rather than handed off to a plugin such as the Flash player. Maybe your downloader isn't compatible with this new feature? If you want to block Firefox from natively playing such files, which basically would return Firefox to the previous behavior, you can switch a setting under the covers.

(1) In a new tab, type or paste about:config in the address bar and press Enter. Click the button promising to be careful.

(2) In the filter box, type or paste media and pause while the list is filtered

(3) Double-click media.windows-media-foundation.enabled to switch its value from true to false. Not sure whether this takes effect immediately when you reload the tab you were in or whether you need to restart Firefox before it takes effect.

Does that work for the files you want to download?

Thanks but I'm an amateur when it cones to computers and I am not comfortable making changes like this.

My problem seems to be mostly with Youtube. I know that in the past, downloads of videos that were part of a play list never work. It has to be a single movie by itself for it to work. The failures I have been experiencing weren't identified as part of a play list but that's the same results I got when they are. I just now downloaded a "Flip the frog" cartoon on Youtube and it downloaded perfect. Do you know if Youtube has made any changes in the last several days?
 Thanks for all your help. I think Firefox has the best help forums of anyone. I can only wish that MSN would copy Firefox's style. MSN has  the worst help forum I have ever experienced!!!

The about:config screen initially shows a zillion settings, but when you use the filter box and know what you're looking for, it's pretty manageable. Anyway, if the immediate crisis has passed, you could just keep it for future reference.

I don't know if you've seen it, but YouTube has the option to view Flash only, or split your viewing between HTML5 and Flash viewers. If you ever opted into that HTML5/Flash combination, it might explain inconsistent behavior on the site. To check/change your setting, try this page:

https://www.youtube.com/html5

I had the problem.Try using Seamonkey browser,Its a lot like Firefox.It took care of the problem for me.

I just found another way.

(On 32-bit Windows) C:\Program Files\Mozilla Firefox

   Go to the Windows Start menu and click on "Computer".
   In the Explorer window that opens, double click Local Disk (C:) to open the C:\ drive.
   Find the "Program Files (x86)" folder or "Program Files" folder.
       On 32-bit Windows, double-click the Program Files folder to open it.
       On 64-bit Windows, you will see a "Program Files (x86)" folder AND a "Program Files" folder. Open the Program Files (x86) folder. 
   Look for a Mozilla Firefox folder.Once you find it, delete it.I found 3 and deleted all of them. Now, go ahead and reinstall Firefox.

You probably have your problem solved by now, but if not hope you check back here these should work!