Check the date and time and time zone in the clock on your computer: (double) click the clock icon on the Windows Taskbar.

Check out why the site is untrusted and click "Technical Details to expand this section.
If the certificate is not trusted because no issuer chain was provided (sec_error_unknown_issuer) then see if you can install this intermediate certificate from another source.

You can retrieve the certificate and check details like who issued certificates and expiration dates of certificates.

• Click the link at the bottom of the error page: "I Understand the Risks"

Let Firefox retrieve the certificate: "Add Exception" -> "Get Certificate".

• Click the "View..." button and inspect the certificate and check who is the issuer of the certificate.

You can see more Details like intermediate certificates that are used in the Details pane.

If "I Understand the Risks" is missing then this page may be opened in an (i)frame and in that case try the right-click context menu and use "This Frame: Open Frame in New Tab".

• Note that some firewalls monitor (secure) connections and that programs like Sendori or FiddlerRoot can intercept connections and send their own certificate instead of the website's certificate.
• Note that it is not recommended to add a permanent exception in cases like this, so only use it to inspect the certificate.

The clock is set to internet time so is always correct. The rest of the answer doesn't solve the problem of why all the https (secure) sites have stopped working at the same time. The technical reason is that the certificate is not trusted because no issuer chain was provided (sec_error_unknown_issuer), Getting new ones from multiple sites should not be necessary and anyway they work on my other browsers for sites such as Google, Facebook and Linkedin, so I suspect it must be a Firefox problem. Induced by what though?

You need to check who is the issuer of the certificate. That is the only way to find what is wrong in your case.

I eventually solved the Firefox problem by doing a system restore using a restore date four weeks before the Windows update that appears to be the causal event. Later the Windows updated again (automatically) and didn't cause any problems with Firefox which is now working normally. However, Chrome then stop working, which I fixed by completely uninstalling an Ask Toolbar that installed on that date for some reason (temporary use of IE related maybe?).

As they say, If you have any other account on your computer please change the time and date. It will work fine again. The date and time need to be correct or else your internet connection will have this kind of problem.

Keep in mind if you are upgrading to Firefox 32 any sites using 1024-bits certificates will show the warning.