What is MVC?
MVC is an approach to separating your code depending on what role it plays in your application. In the application flow it starts with a controller that is loaded. That Controller executes a method which retrieves data using Models. Once it is done the controller decides what View to load, which contains the output your visitors get to see.
Fuel's routing decides based on the requested URL what controller to load and what method to call upon it. This is where your application starts working. The Controller decides what actions to take, what to do with any user input, what data gets manipulated and which View is shown to the user. The Controller does none of these things itself however; it calls upon Models and Classes to do the work.
Read more about Controllers.
Whenever data needs to be retrieved, manipulated or deleted this should always be done by a model. A Model is a representation of some kind of data and has the methods to change them. For example: you never put SQL queries in a Controller, those are put in the Model and the Controller will call upon the Model to execute the queries. This way if your database changes you won't need to change all your Controllers but just the Model that acts upon it.
Read more about Models.
Views contain your HTML, which should never be found in your Controllers or any other class that is not
specifically meant to create output. By separating your layout from your logic you ensure that when you decide
to change your layout you only have to change the views and won't have to care about the Controllers.
Views should thus contain little more than echo and foreach usage of PHP.
Read more about Views.
Once your application gets more complex you'll discover that it gets hard to decide if a piece of logic really belongs in the Controller, what if it is very specifically about the View and has little to do with your application logic? This is where ViewModels come in; they are the glue between your controllers and your views.
Read more about ViewModels.