I closed the web browser which is Internet Explorer 8.0, I then cleared my cache, cookies and stored passwords. Still I was getting the error The Local Security Authority cannot be contacted. I then decided to clear out my Windows temp files in the system location and the %APPDATA% location. I rebooted my machine. After the reboot I brought up my web browser and went to the website and still got the error The Local Security Authority cannot be contacted.
After all this I finally decided to look in the control panel, I am running Windows 7 which has the control panel applet called the Credential Manager. Once I opened the Credential Manager I looked under Windows credentials and found the site listed as this is an internal SharePoint website which used internalmachinename.domain.com as the domain. Once I found the site I clicked on Remove from vault. Once I clicked remove from vault the error went away.
So to summarize:
To get rid of the error The Local Security Authority Cannot Be Contacted when trying to reach a SharePoint or domain authentication type website check that your credentials are not stored in the Credential Manager in the control panel.
I hope this helps someone out as it cost me too much time!!!