The most common forms of heart disease in adult cats are those affecting the muscle of the heart itself (cardiomyopathy):

Hypertrophic cardiomyopathy
The most common form of cardiomyopathy is hypertrophic cardiomyopathy, in which the heart muscle becomes abnormally thick which prevents the heart from working properly and reduces the amount of blood flowing through it. Hypertrophic cardiomyopathy can be caused by an increased workload for the heart. If the heart has to do more work to pump blood out then the muscles in the heart get bigger (just like a weight-lifter's muscles get bigger when they work-out a gym). However, hypertrophic cardiomyopathy can also occur in otherwise healthy animals and the exact cause is often unknown. It is more common in certain breeds, eg Persians, which suggest that it may be inherited in some cases.

Dilated cardiomyopathy (DCM)
This disease is caused by the stretching of the heart muscle walls so that the heart swells (like a balloon filled with water). The contractions of the heart muscle become very weak so blood is not pumped around the body effectively. This is often seen in cats whose diet contains insufficient amounts of a chemical called taurine. However, DCM is much less common now because pet food manufacturers add extra taurine to their cat foods. In humans, heart disease is usually the result of damage to the heart muscle caused by blood clots (myocardial infarction) - this causes the signs of a heart attack. However, cats do not get this kind of heart disease.