CFormatter is one of Yii's core components and used to format values displayed in widgets like CGridView or CDetailView. All classic formats like numbers, text, urls or datetime are supported but you may have to use a formatting rule that isn't built in. Let's say we want to display arrays of strings (like tags associated with a product model). When using CDetailView we would usually make use of PHPs implode function to create a simple string value of all values in the $product->tags array.
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()'.
Yii makes it really easy for all to use their already made objects that automate everything we do. It provides also great power of flexibility and styling but hey, we programmers tend to complicate our lives and push a little more the power of our tools.