Abnormalities may affect all aspects of the heart.
The heart is a muscle, and as such, abnormalities of heart muscle may occur, including abnormal thickening, weakening of the squeezing strength, as well as stiffening of the muscle.
The heart is an electrical organ, so arrhythmias (abnormalities of heart rhythm, and hence abnormality of the electrical system) may affect the top of the heart--relatively commonly--causing atrial fibrillation, for example, while arrhythmias of the bottom chambers of the heart can be more risky, causing ventricular tachycardia or fibrillation.
The heart has valves, and commonly minor abnormalities of heart valves are found. These may lead to heart murmurs on physical exam (nothing more than a sound the physician hears when they listen to your heart); more importantly, however, if valvular heart disease is advanced, it may lead to severe leaking or narrowing of the valve, causing symptoms of congestion, chest pain, dizziness, or heart dysfunction.
The heart has blood vessels to feed all the structures, and it is commonly known that cholesterol buildup may cause narrowing of these blood vessels. Coronary artery disease is common, and may lead to a heart attack, particularly if the disease abruptly ruptures open and blocks the artery acutely. Coronary disease may be well-prevented with control of risk factors such as blood pressure, cholesterol, obesity, or blood sugar levels, and with exercise.
Other causes of heart disease include abnormalities of the great vessels (such as aortic aneurysms), the lining outside of the heart (the pericardium), and can also include congenital lesions present at birth such as holes in the wall of the heart.