In a normally configured web server,
www.website.com/index.php
tells it to load the index.php file from the webroot directory. This
www.website.com/index/
tells it to look in the /index/ directory for a file matching what is listed in the DirectoryIndex configuration directive in the server's httpd.conf file or in an .htaccess file in the web site's webroot directory (see http://httpd.apache.org/docs/2.2/mod/mod_dir.html#directoryindex for details).
A sample configuration directive would look like this:
DirectoryIndex index.php index.html default.html default.htm
This would tell the web server to, when no filename is given, to look in the current directory for an index.php file, if not found look for an index.html file, and if not found a default.html file and so on for everything else on the line.
If you want to be able to run PHP files without having to have .php on the end of them, you would set a default handler in your configuration files (httpd.conf or .htaccess) such as:
ForceType application/x-httpd-php
This will tell Apache to handle all files as PHP whether they have a .php extension or not. (See http://httpd.apache.org/docs/2.2/mod/core.html#forcetype )
You can also use ModRewrite to remap the URLs if you prefer, but ModRewrite directives can be confusing and difficult to troubleshoot.