Right-click the extension download link (in Mozilla Add-ons, where it says "Download Now"), select "Save Link As..." and download/SAVE the .xpi file to your computer in a location where you can find it later for step 1 or 2 below. Then you can do one of the following:

1. You can then manually install it by dragging the file onto an open Firefox window, OR
2. Open the Add-ons window (Tools > Add-ons > Extensions), click the "Install" button at the bottom left, navigate to the location of the downloaded file, click/double-click on the .xpi file to install. Edited by Helper7677; Install button is an option in an extension that Helper7677 has installed

See: http://support.mozilla.com/en-US/kb/unable%20to%20install%20add-ons#w_you-are-asked-to-download-the-add-on-rather-than-installing-it

• Note: you can do this even when not "asked to download rather than install"

Other issues needing your attention

The information submitted with your question indicates that you have out of date plugins with known security and stability issues that should be updated. To see the plugins submitted with your question, click "More system details..." to the right of your original question post. You can also see your plugins from the Firefox menu, Tools > Add-ons > Plugins.

• Shockwave Flash 10.1 r102
  • Security update version 10.2.152.26 released 2011-02-08

1. Check your plugin versions: http://www.mozilla.com/en-US/plugincheck/
   • Note: plugin check page does not have information on all plugin versions
2. Update the Flash plugin to the latest version.
   • Download and SAVE to your Desktop so you can find the installer later
   • If you do not have the current version, click on the "Player Download Center" link on the "Download and information" or "Download Manual installers" below
   • After download is complete, exit Firefox
   • Click on the installer you just downloaded and install
     • Windows 7 and Vista: may need to right-click the installer and choose "Run as Administrator"
   • Start Firefox and check your version again or test the installation by going back to the download link below
   • Download and information: http://www.adobe.com/software/flash/about/
     • Use Firefox to go to the above site to update the Firefox plugin (will also install plugin for most other browsers; except IE)
     • Use IE to go to the above site to update the IE ActiveX
   • Download Manual installers.
     • http://kb2.adobe.com/cps/191/tn_19166.html#main_ManualInstaller
     • Note separate links for:
       • Plugin for Firefox and most other browsers
       • ActiveX for IE

Thank you very much. Your method 1 works, but sounds non-trivial. I want to try method 2, but I find that the "Tools > Add-ons > Extensions" window does not have an "Install" button at the bottom left, only a "Find Updates" button at the bottom right. What should I do then ?

You are right, there is no "Install" button. Select the .xpi file and drag it to the Firefox window. This method works for both Firefox 3.6 and 4.0. Then Firefox will ask for permission.

Sorry. You are correct; there is no "Install" button on the Add-ons window. I have an extension that has an option to show that button.

I need to install an "unadorned" copy of Firefox so that I do not encounter this kind of problem again. Thanks for pointing out my mistake.