Controllers are just Ruby Class files which have a series of instance methods inside
Basic Explanation
Rails controllers are basically files where actions (methods) are kept
Each time you access a Rails app, you're sending a request to the system. The various technologies inside Rails route that request to a certain action, where your code can use the passed data to perform some sort of action (hence the name). The actions are kept inside controllers to give the application structure 
So if you access http://yourapp.com/users/new, it tells Rails to load the new method in the users controller. You can have as many actions in the controllers as you want, but you have to tell the Rails routes system they are there, otherwise they won't be accessible
Proper Explanation
Rails Controllers are just Ruby Classes, storing a series of actions
The "actions" (instance methods) work on passed data (params) to create objects that can either be passed to the model, or used inside other methods
Whenever you send a request to Rails (access a URL), it first uses the ActionDispatch middleware to send your request to the correct Class (controller) instance method (action), and then your code does something with that data
Your job as a dev is to connect the right controllers with the right models, presenting the right data the user at the right time