Dogs (as well as other pets like cats) have a unique structure called the third eyelid, a thin membrane that protrudes from the lower inside portion of the eye and swipes tears across the surface. Behind the third eyelid are tear and lymphatic glands. Periodically one of these glands may protrude over the top of the third eyelid, forming a red bulge. Vets call this condition cherry eye, and it can cause heavy tearing and frequent blinking.