Cat Panting - Photo copyright Kevifornia (Flickr)
Have you seen a cat panting? My late and sweet lady cat, Missie, used to pant when she was stressed going to the vet. She did not like being inside a car and became a bit agitated and started to pant (sometimes). I did see her pant sometimes when she was relaxed and the ambient temperature was hot.
When it is hot, cats do sometimes pant just like dogs. It seems to be a personality trait. Some cats are more prone to it than others. When you think about it, it is surprising that cats don't pant more often. They have lots of fur with a reduced possibility of cooling through sweating. They must get hot in the summer. Black cats also absorb heat so will get hotter.
Panting cools dogs and cats down through the process of the latent heat of cooling. It works in the same way as the evaporation of sweat. When saliva evaporates the evaporated water carries the heat inside the body away from the body in the water vapor. Also the water in lungs, mouth and on the tongue evaporates cooling the cat. The process is called "thermoregulation" - regulating the internal temperature of the body within limits while the external temperature varies. The sweat glands on the feet don't do much cooling.
Cats also cool themselves in the same way by licking themselves. The saliva deposited on the fur evaporates cooling the cat.
This video below, shows a young cat panting. He is panting because he is doing a lot of playing fetch like a dog. Also the ambient temperature might be quite high.
A cat biting his tail and panting may just be play or a form of OCD - obsessive compulsive disorder, brought on, perhaps, through stress.
A cat in shock might pant too. "Shock" in medical terms is caused by insufficient blood flow and oxygen to meet the cat's requirements.
In the USA a bite from one of the pit viper snakes - rattlesnake, cottonmouth and copperhead, will cause panting with other symptoms.
Finally, one symptom of hyperthyroidism is panting.