Sorry for the late reply
Include will attempt to include a file you specify. If it doesn't exist, the script will continue, giving a warning.
Require will do the same thing, but if the file doesn't exist, it will halt the script.
The _once modifiers mean that it will attempt to include the file only if it hasn't already been included. The difference between require_once and include_once is the same as noted above.
Cheers