Instead ofhttp://yoursite.com/login, tryhttp://yoursite.com/index.php/login...

What may have happened is that you had pretty urls on, but when you move a site up or down a folder, the .htaccess file gets missed (as it's often hidden), or it's not edited to refer to the correct document root.

After logging in you could try visiting /index.php/dashboard/system/seo/urls, and there turn off and on again pretty urls to regenerate the .htaccess file.

Failing that, check to see if you didn't move the existing .htaccess file. If you didn't, move it in place and edit it. Inside there will be a line starting with RewriteBase.. if you are running a site from the root directory, that line needs to look like:
RewriteBase /

Pretty URLs have never been turned on.

Here is my current .htaccess:
# Use PHPBETA Single php.ini as default
AddHandler application/x-httpd-phpbetas .php
# Changed PHP handler from application/x-httpd-php5s to application/x-httpd-phpbetas on Wed Nov 4 16:38:38 MST 2015.
<IfModule mod_rewrite.c>
RewriteEngine On
RewriteBase /
RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME}/index.html !-f
RewriteCond %{REQUEST_FILENAME}/index.php !-f
RewriteRule . index.php [L]
</IfModule>

It looks okay to me.

How does the server process the URL? How does it know what to do with the "/Login" that is appended to index.php?

I gave up and reinstalled. Unfortunately, I lost data, and will have to spend many hours rebuilding it. I tried a system restore, but that wasn't successful either.

Thanks for helping.

By the way, I noticed that the tmp folder and the php.ini file must reside in the root; in my zeal I deleted both of them. Restoring them didn't solve the problem, so who knows what else I might have inadvertently deleted.