Animation 101: How does an Animation component work with a hierarchy?

I'm just getting my feet wet in Unity and I'm trying to figure out the standard 3rd Person Controller, the construction worker. I just want to swap out the construction worker for my own character, which has two basic test animations. I have the Animation Component activated and set to 'Play Automatically,' just like with the default construction worker, but my character moves around as stiff as a board. The animations aren't being loaded at all.

The ONLY time I've gotten animations to work at all is when I load the Animation Component right on the object itself. This 3rd Person Controller, however, loads animations in the object's parent: the controller object.

What am I missing? How does the Animation Component work within a hierarchy?