What is Aperture?
Simply put, it's the opening in the lens. It is referred to as the f/number. The larger the f/number, the less light comes through the lens. The smaller the f/number, the more light comes in. I know it seems reversed but it has to do with more complex focal length calculations (not getting into it!).
What is Depth of Field?
The Depth of field (DOF) is closely related to the aperture. It is the amount of the photo that is in focus. A large DOF will have most the image in focus and a small (or shallow) DOF will only have a part of the image in sharp focus.
Now get out there and experiment with different f/numbers and DOF!