Paul Miller in Scientific American:
The inner eyelid of cats–more properly called the palpebra tertia but also known as the nictitating membrane, third eyelid or “haw”–has been regarded by some as a biological curiosity much like the human appendix or wisdom teeth. In fact, some veterinary articles in the early 1900s describe methods for removing this supposedly irrelevant structure so as to facilitate examination of the eye. Despite these perceptions, the third eyelid of cats plays an important role in maintaining the health of their eye surface. In fact, it is so important that among mammals and birds the norm is for a species to have a third eyelid and those lacking one–such as humans and some of our fellow primates–are the true oddities in nature.
The anatomy of the third eyelid is complex. It is a fold of tissue covered by a specialized mucous membrane (the conjunctiva) that faces the inner surface of the eyelids (palpebral surface) on one side and the cornea on the other side (bulbar surface). Embedded in the bulbar surface is a dense population of lymphoid follicles that are in contact with the surface of the eye and the tear film, a thin layer of liquid. These structures function as the lymph nodes of the eye, trapping unwanted dirt and detritus.