When teachers say "depth," they mean analysis. You should challenge them to articulate themselves better, like they challenge you. They should add depth to their feedback, instead of being so general.
What they want is hard to explain. A superficial essay just tells about a book or film or whatever you are writing about. A deep essay says something unique, something that others might even disagree with, something particular to you -- and it shows that you are being affected by the subject.
As an example of superficial and deep writing, read a few movie reviews online. Some reviews just tell about the movie, while others challenge it, critique it, consider the implications. You can tell what is deep.
Kevin, EssayForum.com