"Breathing" is when you change the focus on a lens the field of view changes. Different types of lenses have different levels of breathing. For example: most lower cost zoom lenses (under 10k) generally have moderate to extreme changes to the field of view when racking focus. Also, many low cost zoom lenses made for still cameras usually have breathing issues.

This is one of the nice things about using PL lenses on a camera if you can. Good PL lenses have been engineered to eliminate most breathing, this is part of the reason that PL lenses are so expensive.

To try out your own lenses to see the breathing, all you have to do is frame up your camera so you have objects that are near the edge of the frame. Then rack your lens going from infinity to near and back again. If your lens has breathing issues you will see that the objects on the edge of the frame may be moved out of the frame. To me it looks sort of like a small zooming motion.

Usually "breathing" is only really noticeable when doing an extreme rack with a non moving camera. One way to get around it is to add a little motion with the camera along with your rack focus. This changes the cameras view point and helps distract the viewer from mechanics of the focus being pulled.