You would use them to include another file. A common example is to use include_once('header.php') on each page, so you just have to maintain one header file (Makes it easier to add/remove links).
They both essentially do the same thing but if you use require, the file HAS to be available or you receive a fatal error whereas include_once will try to include it but if it can't find it the page will still render.
In practice, if you're trying to include a file which doesn't exist then you're probably doing something wrong. In most cases, you'll want to use Require.
Bookmarks