Difference between require() and require_once(): require() includes and evaluates a specific file, while require_once() does that only if it has not been included before (on the same page).
So, require_once() is recommended to use when you want to include a file where you have a lot of functions for example. This way you make sure you don't include the file more times and you will not get the "function re-declared" error.
Difference between require() and include() is that require() produces a FATAL ERROR if the file you want to include is not found, while include() only produces a WARNING.
There is also include_once() which is the same as include(), but the difference between them is the same as the difference between require() and require_once().