Hello George,
Based on your description, I understand that the issue occurred when you try to use Open with Explorer in your SharePoint Online library.
You can follow the steps as below to troubleshoot this issue.
1. I suggest you use 32bit Internet Explorer Version 9.
2. Run desktop for Office 365 by following http://onlinehelp.microsoft.com/en-us/office365-enterprises/ff637594.aspx
3. Make sure that the WebClient service is running.
• Click Start, click Run, type services.msc, and then press Enter.
• Locate the WebClient service, and check if its status is set to Started.
If not, double-click it, click Start, and then click OK.
4. If the issue persists, you can add https://*.sharepoint.com/ in the Trusted Sites zone of Internet Explorer.
For your reference:
Software requirements for using the Open with Explorer command
How to troubleshoot issues with "Open with Explorer" in SharePoint Online for Office 365
If the issue persists, the screenshots and any information related would be appreciated for our further research.
Thanks,
Cherry Wang