What is the aperture on the camera?

I don't understand WHAT the aperture is on my camera (Nikon d5100) as well as the role it plays in exposure, shutter speed and depth-of field, and HOW it can be adjusted.

My understanding is that the aperture limits the amount of light passing through the lens, where a smaller aperture lets in limited light, and a large aperture lets in lots of light. Please correct me if I'm wrong and go into more detail.

Nope, you've got it. The aperture in your lens is a hole, the dimensions of which can be adjusted to control the amount of light entering the camera.

Aperture does let in more or less light depending on the size (f-ratio) It also affects the depth of field in focus - small aperture, large depth of field in focus, large aperture, shallow depth of field in focus. Obviously with varying light due to changes in aperture you'll need to adjust shutter speed to correctly expose the frame.