Every now and then I run into this concept in different contexts: animation controllers, camera controllers, AI controllers, character controllers, etc. All these sound like very different components, yet the term "controller" is applied to all of them. Could someone please explain the basic idea behind this concept? Thanks.

A good starting point:

http://c2.com/cgi/wiki?WhatsaControllerAnyway

It generally depends on whom you ask. Sometimes, the controller is seen as the mediator between the model and the view in MVC, sometimes it's the "read input" part of MVC, and sometimes it's just a class which is responsible for applying operations to a dumb class (for instance, a camera probably implements only simple primitive operations, because complex behaviors are better implemented in distinct classes).