Recurring password-protected popups on older Macs

I know I have already asked this question but have only received quotes from some very old posts, which I couldn't understand, or which sounded too draconian. The question at hand is the continuing problem I have on my older Macs. At any and all times popup windows appear which will not close until I have entered my computer's password. They reference some utility or required execute file and will not close until I enter that password. This is very disruptive. It started on my Mac mini several years ago. It spread to my iMac and to my hard drive backups. I am afraid it will start on my restored MacBook Pro, which is on the network with all of my Apple products. I dare not download any files stored by Time Machine. I ran Malware Bites and when that didn't work I was told by them that this particular virus was immune, because the Apple version wouldn't work. Of course it wouldn't. I took the Mini into the Apple Store. The Genius I saw didn't recognize this particular virus, but asked me to leave it with him. I got a call about a week later to pick up my computer. I was pleased, until I arrived to find the computer had been wiped back to factory settings because no resolution was found! So my only choice if I wanted to stay at the level I had been working on was to upload files from my hard drive, and that brought the whole thing back! Someone out there seems to know about what causes this stupid situation. I would love an explanation and solution that works, and doesn't involve killing off too much information required by my computers to operate successfully. Help!

[Re-Titled by Moderator]