<?php require_once('../Connections/localhost.php'); ?>
The ../ would indicate that the connections folder is not in the current folder, but one folder up. So if you had the page with this include in a sub directory in the site called subdir it would indicate that the connections folder is not in this subdir folder but instead is in the folder above it.
The trick to resolving errors that occur for the links like this is to determine if the path is correctly pointing to the file, if it is and you still get the error it indicates that the file has not been uploaded to that location on the live server. Here is a link to a quick page that discusses the types of links for a web page:
Thanks, the article is very helpful, but I am afraid that we would need to both be looking at the files, and possibly the file structure of on the remote server. I have these in a zip file that I will upload in the trouble ticket under this same title...