Coronary artery disease (CAD; sometimes called coronary heart disease, or CHD) is the most common kind of heart disease. It happens when the arteries that feed blood to the heart muscles harden and get clogged through atherosclerosis. This process starves the heart of the oxygen and nutrients it needs to function well. Worsening CAD can lead to permanent heart damage and ultimately death. 
Damage to the coronary begins early in life, and is usually without major symptoms until things get pretty bad. At that point, one of the main symptoms of CAD is chest pain (angina) that lasts on the order of minutes (as opposed to hours or seconds). However, the specifics of how this feels, when it occurs, and what’s associated with the pain depend on the specifics of the clinical disease; heart attacks (where a clot breaks off from a ruptured athrosclerotic plaque and blocks a narrower downstream artery) are different from, say, stable ischemic heart disease (where pain appears upon exertion and settles down afterwards).
The specifics of how chest pain feels, along with risk factors such as high cholesterol, high blood pressure, diabetes, lifestyle factors such as smoking, and a family history of cardiovascular diseases is often enough to raise a strong suspicion of CAD. Physical examination and additional testing by a medical professional along with results along with blood work can provide further evidence.
The specifics of medical treatment ultimately depends on the specifics of the person and the clinical disease they have. The treatments tend to fall into a few categories:
Medically treating underlying risk factors to slow CAD progression. For instance, statins can be used to lower cholesterol - and prevent further heart damage
Drugs for symptom relief. For example, nitroglycerin can be used in people with stable ischemic heart disease to reduce short-term chest pain. Drugs that work on the heart like beta-blockers and calcium-channel blockers are often used to control pain and discomfort in the longer term, and also may help improve other outcomes in people who have established clinical disease
In severe cases, invasive procedures like surgeries are sometimes used
Medical treatments are ideally added on top of lifestyle changes that can help slow CAD progression. Things like quitting smoking and exercise can help a lot.
Yes, many supplements have been studied for both preventing clinical CAD as well as in people who already have CAD. Given that high blood pressure and high cholesterol cause CAD, supplements that affect those outcomes may also affect the risk of CAD progression.
There’s a big connection, although there’s still some controversy around the details, such as whether or not limiting saturated fat without considering what it’s being replaced with will do much. Just like with atherosclerosis, guidelines suggest that a diet abundant in vegetables, fruits, legumes, nuts, whole grains, can help reduce the risk of CAD. Weight loss for people with higher BMIs may also play a role in preventing CAD, although the current evidence suggests that modest weight loss (in the 5-10 kg range) may have limited benefits.
Exercise can play a big role in preventing CAD and also can help people who have CAD improve their health. Guidelines suggest that the role of resistance training in preventing CAD is unclear, but resistance training is still encouraged.
Atherosclerosis of the coronary arteries leads to CAD.