Let's break your question in two:
- What is the role of Controller in the MVC flavour offered by Monorail and ASP.NET MVC?
- How url mappings relate to applicative actions?
My take on 1:
As this type of question lend itself to many religiously answers, I believe that the is no "one way to rule them all".
Now in the Monorail and ASP.NET MVC (and also RoR of course), a Controller is simply a collection of Actions.
The correct question then is "What is the role of Action"?
In my book (the unwritten Monorail-in-action book ... :) ), the role of the Action is to separate the Domain Model from the presentation, both in terms of data structures, and in concerns. Everything that is specialized to the fact that the interface with the domain is through WEB requests, is the controller's layer responsibility. That includes data binding and transformations, dealing with Authentication (but not authorization), and making decisions for the view templates.
So, an action will take parameters from the incoming request (a web is not a Domain concern), bind these to a meaningful data that can be send to the Domain as a Query or Command, in the Domain's language, without no cookies, FORM, QueryString, and other "web stuff".
It would also, when viewing data, will transform the domain objects that got returned from the Model, into a View Model, that in the same book mentioned earlier is a model separated from the Domain model, and is in charge of supplying the view-template with all the data and the decision making it needs. So, for eg., the view should not ask if (view.User.IsAdmin)
and render an "EDIT" button, but instead the Controller's action will have made this question, and supplied the view with a decision, for the view to ask if (view.ShouldRenderEditButton)
So, the Controllers layer separates WEB concerns from DOMAIN concerns.
As for question no. 2:
The idea of mapping the url as Controller/Action is simply a consequence of taking the "Convention over Configuration" approach. Meaning, it would be easier for developers (and consumers) to work with a schema that is common across different web applications.
Having said that, it is not written in stone, and like any Convention, it is a basis for Adaptation. So if you are building a website and the product manager asks for "pretty urls", then you just set up your routing engine accordingly.