usually 401 Unauthorized [w3.org] is the correct response for a failed authentication request.
upon receiving such a response it would be normal behavior for the user agent to collect the authentication information and make a new request while providing an authentication header.
perhaps it is a cache issue.
try a different user agent or clearing cache and a new request.

here is the protocol for Basic and Digest Access Authentication [ietf.org] which may have some useful information for you.
you might also want to read up on the correct use of the http Authorization [w3.org] header for clues.

try a different user agent or clearing cache and a new request.

Everyone who goes to my site is getting this error. I am not sure what you mean by that.

This is something that all the sudden happened to my site and I had no control over.

I think I got hacked...nothing else seems to explain this.

I'd pursue this

"they forced a permission check on the server"

and find out what they mean. It could be your host has closed access to this site for one reason or another, possibly, as you say, someone has hacked it or some other reason.

Trying to directly access a page that you do not want showing up, such as your images folder, will generate this problem too. If it's happening sitewide check your .htaccess settings.

that's a good point, JS_Harris.
if you have Options -Indexes [httpd.apache.org] that will forbid a directory listing unless there is a valid resource available such as one specified by the DirectoryIndex Directive [httpd.apache.org].
this is all assuming an apache server, of course.