As written on the API docs: An Action Provider Widget is a widget that has declared several actions by using its 'public static function actions()'. After, a Controller can easily import all its declared actions on its 'public function actions()'.
More or less I use auto-complete field for a form. CJuiAutoComplete is a yii widget for this purposes.
The common use case is to make an ajax request and retrieve a list from the database. So in this case you have to write a controller action to return you a json encoded list.
What if I had a generic action to do this work for me?