If your cat is healthy, it will normally have a good appetite, relatively active mood, clear and clean eyes, and a slightly wet nose. Signs of an illness in cats can be divided into two groups: apparent and unapparent ones. A quick summary for each of the groups is represented below.

Unapparent signs of cat illness

If you have noticed one or several of these and they have become continual, it will be best to have your feline checked by a specialist.
Behaviour changes. Has your usually affectionate cat become moody or unfriendly? Does it avoid being petted without an obvious reason? Has your usually unfriendly cat become more social and started to follow you everywhere? Even the slightest change in the usual cat behaviour may indicate possible health problems.
Appearance changes. Has your kitty lost weight lately? Has it unexpectedly gained weight while the diet and appetite haven't changed? Does its hair shine less or look thinner? Has it experienced excessive hair loss lately? Any of these signs can be early warnings there's something wrong happening with your cat. They can be age changes, but even if it's so, it means your cat has become a senior, therefore its needs have to be revised accordingly.
Appetite changes. Cats are known to be finicky eaters. Their appetite can change from day to day, but if the tendency gets apparent, this is a reason to consider visiting a vet. For example, your cat has become ravenously hungry or has lost any interest in food, and such behaviour persists over several days. If your cat is thirsty all the time, this can be a sign of kidney problems or diabetes.
Vomiting. Some cats (mostly, long haired ones) cough up hair balls which you can easily identify. This can be minimized by regular grooming. However, any prolonged and excessive vomiting should become a cause for concern.

Apparent signs of cat illness

If you cat has any of the signs listed below, it's recommended that you take it to a veterinarian as soon as possible.
Diarrhea. We all have stomach disorders sometimes, and they don't necessarily mean serious health problems. However, don't ignore diarrhea that persists for several days in a row.
Constipation or urination problems. These are easier to notice if your cat is indoor and uses a litter box. Has it started spending there more time without passing feces (urine)? Has it begun to cry trying to ease itself? These signs can indicate a problem. In addition, a refusal to use its litter box without an apparent reason can also indicate a health problem.
Blood in feces or urine. If there's blood in your cat's urine or feces, take it to a vet as soon as possible. Possible causes of blood are infections, inner injuries caused by foreign bodies in the stomach or intestines, coagulopathies (bleeding disorders), urinary tract stones, poisoning (especially from rat poison products), metabolic and other diseases, including cancer.
Abdominal distension. It can be caused by a roundworm infection, enlargement of any abdominal organ (including the liver, kidneys, or spleen), internal hemorrhage (bleeding), accumulation of urine from a tear in the urinary tract, and others. If you have noticed that your cat's abdomen is distended (unless it is caused by pregnancy), show your feline to a vet stright away because it may need urgent medical help.
Excessive scratching. It can be caused by an allergy or poor skin condition. If your cat scratches its ears, it can be a sign of ear mites. Check your cat's hair for any signs of fleas, rashes or bald patches on a regular basis.
Discharge from nose, ears or eyes. Just like humans, cats can wake up with some "sleepers" in their eyes. If you spot unusual thick, watery, or bloody discharge from your cat's eyes, nose or ears, it can indicate a health problem. Unusual discharge may be developing slowly or start suddenly. Show your cat to a veterinarian as soon as possible.
Unusual state of gums and teeth. Check your cat's teeth and gums on a regular basis. If the gums are very pink and your kitty has a bad breath, this may indicate a disease such as gingivitis.
Excessive sneezing. Like people, cats sneeze if something has irritated the nasal passages. If sneezing has become too frequent and continual, it can indicate a problem such as cat flu or an allergy.

Do not forget that everything listed here does not necessary mean that your cat has serious health problems. In most cases, these symptoms are warnings from your little friend that you should not ignore. Take your cat to a vet when there's something worrying about its health. Do not panic: it is always better and easier to prevent a disease than to try to cure it afterwards. It's better to be safe than sorry.

See also: Health problems in older cats