Breathing refers to when a lens' optics change the apparent focal length slightly when shifting the mechanical focus. Some (often more expensive) lenses are designed to lessen the degree this effect. Lens breathing does not prevent one from racking focus or following focus with this lens, but it lessens the desirability of any type of focus adjustment, since it noticeably changes the composition of the shot.
Not to be confused with the suction and expulsion of air from within the lens as its internal volume changes.