The infographic says more or less all that I want to say about the feline third eyelid 😃. I’ll add a few details for the sake of completeness as an infographic is a very compressed source of information.

Not always a health issue
Perhaps an important point to remember is that sometimes you will see the third eyelid in your cat when they are perfectly healthy and resting. Another point is that sometimes when your cat is sleeping their eyelids might be half open and you can then, sometimes, see the third eyelid protruding across the eyeball. This is all normal and does not indicate illness.
Seen all the time?
However, if your cat’s third eyelid is visible a lot or perhaps all the time, it is indicative of a background illness. You should seek veterinary advice under those circumstances. The general advice is that if this interesting piece of feline anatomy is visible all the time, then there’s something wrong.
Although not included in the Infographic, when the third eyelid is associated with a bulging eye the causes may include infection in the tissue behind the eyeball (abscess), bleeding behind the eye (haematoma), and tumour.
Key-Gaskell Syndrome is a rare nervous system disorder of unknown cause. Other signs include dilated pupils, constipation, trouble eating and a slow heart rate. The disease is more commonly seen in the UK than in other countries. The treatment for this “syndrome” is “extensive nursing care for weeks to months” according to my veterinary reference book. The prognosis is poor. Most cats develop aspiration pneumonia.
Haw Syndrome is, as mentioned, “rather common”. It is temporary. The cause is uncertain. As mentioned, it normally clears up without treatment within a few months. Your veterinarian might prescribe an eyedrop solution containing 1 or 2% Pilocarpine (USA) to reduce the size of the protrusion if it interferes with your cat’s vision.
Horner’s Syndrome can result from sunken eye, prolapse of the third eyelid and a small pupil. It might occur as a result of an injury to a nerve or a middle ear infection. There is no treatment. The condition may resolve with time.
Below are some more articles on the eye.