Cats are terrific bang-for-the-buck pets. They are graceful, playful, wonderful to touch, and beautiful. You can get truly beautiful cats of any age at any shelter in the United States. Free kittens are frequently offered by private parties who have neglected to neuter their cat or who have rescued an abandoned litter.
Cats are cheaper to feed than dogs, and, if neutered, vaccinated, and kept indoors, usually require little veterinary care. Cats are social, but not nearly as susceptible to separation anxiety as dogs (separation anxiety is usually the cause of incessant barking, garbage strewing, incontinence, etc., that can happen when a dog is left home alone). Cats can be left alone for a weekend if given access to sufficient dry food and water. Being left home during work hours, even long work hours, is not usually a source of great distress to cats if they are well cared for and given attention before and after work. Cats can happily live indoors in a small space and so are very well suited for apartment living.
As Letrisa Miller mentioned, there are multiple studies showing the mental and physical health benefits of pets, including cats. You could read about this in the American Psychological Assocation's The Truth About Cats and Dogs: Pets Are Good for Mental Health of ‘Everyday People’ and read the opinion of the U.S. Center for Disease Control at Health Benefits of Pets. I have an Emotional Support Animal who is a cat. You can read about how she helps me deal with bipolar disorder at How an Untrained Cat Can be an Assistance Animal.
Another thing Letrisa Miller mentioned is allergies. While some people are allergic to cats, one should also consider the studies that indicate that children with pets are less likely to develop all sorts of allergies.
The downside of cats is scratching furniture (not all do it, but it's often an issue, especially with young cats), problems with urine marking (again, not all do it and usually it's a temporary problem if you keep the litter clean and identify and remedy the stressor that is causing the problem), and cleaning the litter box. The latter requires some care because in addition to rabies (for which you of course will have the cat vaccinated) there is another disease of note that one can catch from a cat. It's called toxoplasmosis and it's usually contracted by the person who cleans the litter box. It can cause problems with a fetus, so pregnant women should have someone else clean their cat's litter box. You're more likely to get this disease from your food than from your cat, however:
While many people think handling cats and their litter boxes are the primary ways pregnant women are infected, a new European study on how to prevent the devastating disease . . . shows that up to 60% of all transmissions can be prevented by limiting exposure to inadequately cooked or cured meat(see Take These Precautions to Prevent Toxoplasmosis During Pregnancy)