Object oriented programming and gameobjects

This is not a question about broken code. In school I'm learning about object oriented design, where you have classes which make objects and really delegate most of the work to these objects. If you have a cat, the cat knows its behaviors and how it works, and then you just let it work. This sounds perfectly suited for Unity and coding AI enemies or items, but I don't understand how to tie in the object in the script with the object in the game. What are some good ways to connect, for example, a cat-looking object in my game with the behavior-of-cat programming object?