The include function simply includes a file, and that is all. If the file isn't found a warning is raised, but the script will continue to execute.
Require has very similar functionality to include(). However, if a file is not found a warning will be raised and the script will halt completely.
Include_Once() and Require_Once()
You should use these functions if you do not want to include a file multiple times (which can cause errors with functions and classes).
file1.php includes file2.php and file3.php.
file2.php also includes file3.php
Without the _once() functions file3 will be included twice, but with the _once() function it will only be included once.