Apache version on server: 2.2.26
PHP version on server: 5.5.9
I have a file called admin_config.php in a folder, which has an .htaccess
Paths you give to include or require are paths on the local filesystem. They are not what you see in URLs to access your site. An absolute path starting with / is from the root of the filesystem. In Windows terms, /foo is C:\foo\. Relative paths like foo/bar are relative to the the PATH configuration variable, which depends on how your PATH is set up which also includes which PHP file was invoked.
It's typically not a good idea to use absolute paths, since those are likely different on different systems (as you are experiencing). On your local machine the site may live in C:\core\..., but on the server it'll be running in /var/www/mysite/core/.... PATHs can also be cumbersome to work with. The best is typically to use __DIR__ or __FILE__ magic constants to construct an absolute path relative to the current file (if that made sense):
require __DIR__ . '/some/folder/file.php`;
This includes the file some/folder/file.php relative to the file in which it is written.