The 'Controller' module is "C" in the Model-View-Controller (MVC) architectural pattern.
It can be useful if you want to use MVC in Drupal like it can be done in MVC frameworks (like ZF, Yii and so on).
The 'controller' module allows to use a Page Controllers (Action Controllers) for your pages. The Controller is just class that has set of actions. Each action is a separate method that will be called for specified path (page).
For example if you have the 'foo/bar' path (URI), then you can create a new class called FooBarPageController and myAction method (you can use other names of course). This method will be called for the 'http://host/foo/bar' address.
Important note 1: it is utility module for programmers only.
Important note 2: the Controller object that acts as page handler is not the same as Controller that implements DrupalEntityControllerInterface - they are different things and used for different purposes.