The coronary arteries are responsible for supplying blood to the heart. Like any other organ in your body, you heart needs oxygen to stay alive. In fact, due to the fact that it never stops pumping, the oxygen demand for your heart is quite high. Although there is a constant supply of blood flowing through the chambers of the heart, none of that blood is used by the heart to feed itself.
The heart gets it's oxygen from a series of small arteries which are supplied from small openings in the very beginning of the aorta. Unlike many other arteries, there is considerable variation from person to person in the exact structure of the coronary arteries. For most people, there are two major branches of the coronary arteries, the left and right. Both of these originate at the root of the aorta, just after the aortic valve connecting the left ventricle to the aorta.
The right coronary artery travels between the right atria and ventricle and supplies most of the right ventricle as well as a good portion of the left ventricle. In most people it also serves as the blood supply for a few other interior structures of the heart, such as some of the papillary muscles, which are involved in controlling the valves between the chambers.
The left coronary artery has several major branches as well, including one that is particularly important. The left anterior descending artery is a branch of the left coronary. Known as the LAD, the left anterior descending artery is responsible for supplying oxygen to the left ventricle. The left ventricle is the chamber which is the strongest and is used to pump blood out of the heart and around the rest of the body. If the LAD should become blocked, the left ventricle will stop working and lead to a heart attack and potentially death. This has lead to the LAD being given the somewhat morbid nickname, "The Widowmaker". Nice thought, eh?
It is interesting to note that the coronary arteries fill in a somewhat different fashion than all other arteries in the body. Normally when the left ventricle contracts, pressure builds up in the ventricle until it gets high enough to be ejected from the heart and pushed out in to the aorta, where the blood is then distributed throughout of the rest of the body. However, when the heart is contracting, the coronary arteries are being compressed by the contracting heart muscles themselves, making it impossible for them to get good blood flow. The coronary arteries thus fill while the ventricles are at rest and the pressure in the walls of the heart is at its lowest. They are the only arteries in the body to fill while the heart is in its resting phase.
The coronary arteries have two major weaknesses. First is that they are a type of artery known as an "end artery". This means that there are not a lot of other ways for the blood to get to the tissues being supplied by them. There is almost no redundant circulation as there is in most other parts of the body. In addition, the coronary arteries have a nasty predisposition to become blocked by atherosclerotic plaques. The combination of these two traits is a major reason to heart disease being the number one killer in America. When a coronary artery becomes blocked by enough plaque and crud such that blood flow is no longer sufficient to supply the heart with oxygen, you are in serious trouble. That's essentially what is happening in most heart attacks and a condition known as angina.