Add define('WP_DEBUG', true); to your site's wp-config.php. That will cause errors, warnings, and notices (non-fatal warnings) to print to the screen. Those are the "debug information" so frequently requested.
It is not advisable to have this enabled on a production (publicly accessible) server but if you have to have the debugging information then you have to have to have it.
You can also add define('WP_DEBUG_LOG', true); and the debugging information will be written to a file named /wp-content/debug.log.
You can then add define('WP_DEBUG_DISPLAY', false); to prevent errors from printing to the screen as you can read them from the debug file.
So, in your wp-config.php you'd have:
define('WP_DEBUG', true);
define('WP_DEBUG_LOG', true);
define('WP_DEBUG_DISPLAY', false);
Your server itself may also keep log file. The location and names of these files vary by OS. On Debian Squeeze, Apache's log is by at /var/log/apache2/error.log. On CentOS 6 it is at /var/log/httpd/error_log. Those have much the same information, but you may not have direct access to them depending on your host and hosting type-- shared, vpn, etc. The database server may also keep logs.