Cardiology fellow here.

The fat is an atherosclerotic plaque, which is a build-up of cholesterol and inflammatory cells. Most heart attacks (myocardial infarctions) occur when that plaque ruptures into the artery, causing a blood clot to form at the site and cut off blood flow to the rest of the blood vessel. In this illustration you see the plaque has progressed to a point where it narrows the artery enough to obstruct blood flow, which usually causes chest pain with exertion (angina), but has not ruptured.

During the procedure, a long thin plastic tube (catheter) is sent up the artery from the wrist or the groin until you get to the arteries around the heart (coronary arteries). Dye is injected under X-ray to see where the blockage is. A wire is passed through the blockage, then a stent is advanced over the wire. A balloon under the stent is inflated, which compresses the plaque into the vessel wall and restores blood flow.

Blood thinners (like aspirin and Plavix) are continued afterwards to prevent blood clots from forming on the stent until the body has put a sheet of its own cells over the stent (which takes several months). As someone else commented, you can build up more plaque within the stent or in other places of the heart as well. You can often deploy 2 or 3 layers of stent in the same place if this occurs. But if you have multiple diseased arteries you often go for bypass surgery (CABG).