Dogs and cats can stop eating for a lot of reasons, including fever, pain, or stress. If your notice your pet's appetite has decreased or been absent for more than 24 hours, you should take it to see a vet. While a decreased appetite and lack of activity are very vague symptoms, should they persist, it would be paramount to visit the veterinarian.
Cats typically suffer more dire consequences when they stop eating, and can quickly develop fatty liver, which is a potentially fatal disease. This is especially applicable to overweight cats as excessive accumulation of fat in the liver can cause liver failure. A cat that stops eating should see a vet promptly because fatty liver can be treated.
If your dog stops eating, he could be being fussy about his food, but if this behaviour persists, something could be wrong and action should be taken.
Many things can cause lethargy in pets, including major problems, such as heart disease. A pet whose lethargy cannot be pinned on an obvious reason, such as from a good workout in the park, may need to visit the vet, especially of other symptoms arise, such as change in exercise tolerance, weakness, collapse, or loss of consciousness.