Detecting illness in a cat can be tricky, simply because they can't talk and their bodies are (usually) covered with fur. Because of the difficulties, cat owners should take their pets to the vet at least once a year for a check-up. If the cat is a senior (over 7 years old) or has known health issues, the vet will probably recommend more frequent visits. There are also signs of illness to watch for in between visits to the vet:
1) Litter box issues
If your cat has always been perfect about using the litter box, and then suddenly starts making messes outside the box, there's a problem, perhaps kidney disease. This is especially true once you've ruled out such obvious concerns as cleanliness. (Cats won't use a dirty litter box anymore than a human would use a dirty toilet.)
If the cat appears to be straining to urinate and/or defecate, get her to the vet. If the cat is trying to go to the bathroom and not producing anything, it could have a blockage somewhere, which could be fatal if left untreated. This is especially true if the cat also seems to be in pain, i.e. it cries while trying to go to the bathroom.
2) Unexplained weight gain or loss
If the cat is losing or gaining weight for no obvious reason, it's time to take her to the vet. This is especially true if the weight change seems sudden. Because of their small size, a cat's gaining or losing a single pound is roughly comparable to a human gaining or losing 10 pounds. Some weight loss in older cats is normal as they lose muscle mass, but extreme gauntness can point to something serious like cancer.
Marked weight gain can result in obesity, which will lead to many health problems. An obese cat can develop many of the same health problems that an obese human will, like diabetes or heart disease.
3) Blood in urine, stool, or vomit
Blood in the urine can indicate a urinary tract disorder, especially if accompanied by straining while trying to urinate and/or increased visits to the litter box. Blood in the stool can indicate a variety of illnesses, some relatively minor. Others, like certain parasitic infections, are more serious. Vomiting blood, however, is always a sign of a serious illness. In some cases, the cat will vomit what looks like coffee grounds, and this is actually partially digested blood. A cat vomiting blood needs to be taken to a vet immediately.
4) Diarrhea or constipation
Untreated diarrhea can result in dehydration, which can be fatal. Diarrhea is caused when too much water is expelled with the stool, thus creating a loose or watery stool. Diarrhea is also associated with an increase in frequency of defecation.
With constipation, the cat produces small, hard, and infrequent stools. A cat typically defecates once or twice a day. Constipation is most commonly caused by hairballs, and can lead to weight loss and anorexia. The occasional watery or hard stool won't hurt a healthy cat, but persistent diarrhea or constipation should be treated by a vet.
5) Changes in appetite or drinking habits
If the cat refuses to eat or drink, get her to the vet. Refusal to eat or drink often means the cat is in pain or is otherwise feeling poorly. Increased thirst, especially when accompanied by increased urination, can point to feline diabetes. Increased appetite can also indicate disease.
6) Repeated vomiting
If the cat vomits up the occasional hairball, that is probably normal. If she vomits several times a day, get her to the vet. If she vomits blood, get her to the vet immediately.
7) Mobility problems
Stiffness, limping, and the like indicate problems, especially in a young cat. In an older animal, they can indicate a condition like arthritis. In any event, have the vet examine her, so he can rule out the more serious problems and/or recommend ways of making your cat more comfortable.
8) Behavioral changes
Sudden changes in behavior also indicate trouble. For example, if a normally outgoing cat suddenly starts hiding all the time, this may mean that she's sick or in pain. Similarly, a cat in pain may become aggressive, especially if you unwittingly touch a sore place.
In short, be attentive to your cat's behavior and habits; these are your clues to determine the health and happiness of your feline friend.