A heart attack is a plumbing issue, affecting the fuel supply to the muscle of the heart.
The medical term for a heart attack is a myocardial infarction. Sounds fancy, but like most things in medicine - it's actually quite simple. (Myo-Muscle, Cardial-Heart, Infarct - Dying tissue from lack of blood supply).
As discussed in our general overview, the heart has three main arteries supplying it - one on the right and two on the left - namely the right coronary artery, the left anterior descending artery and the left circumflex artery. A blockage in one of these arteries, or their smaller branches leads to a lack of blood/oxygen supply the muscle. This starts the process of muscle death - leading to the symptoms of a heart attack. 
We all know the possible symptoms - central/left-sided chest pain which often radiates to the jaw or left arm accompanied by nausea or shortness of breath. People often describe the sensation as somebody sitting on their chest. 
Your risk of having a heart attack goes up with age, as well as the usual things your doctor pesters you about such as cholesterol, blood pressure, diabetes, smoking, etc. 
Fortunately, chest pain (even convincing symptoms) isn't always secondary to a true heart attack - so when you arrive in the ED, we have to run some tests. 
So not to overly delay treatment, a paramedic or doctor will often give you 300mg of aspirin before confirmation - it is a blood-thinning medication. 
The next step is an ECG (Electrocardiogram) - a snapshot of the electrical activity in the heart. A heart attack will often lead to disruption of the heart's usual circuits and we look for changes that reflect that - namely the ST segment and the T waves. Thankfully, you don't need to read your own ECG, so we won't go into detail. I only mention the ST segment as whether it's up or down helps us to stratify the severity of your heart attack.

STEMI - ST Segment Elevation Myocardial Infarction
When we see ST elevation on the ECG, coupled with your symptoms, that is enough evidence that a serious heart attack is occurring. You will be given a second blood-thinning agent (such as clopidogrel or ticagrelor), before being rushed to the Cath lab for a coronary angiogram. You will likely be given a third blood-thinner before any procedure (heparin or clexane). 

NSTEMI - Non-ST Segment Elevation Myocardial Infarction
An NSTEMI is usually indicative of a less severe blockage, but it can be quite serious all the same. Doctors sometimes might refer to this type as a minor heart attack. You too will be given aspirin, either clopidogrel or ticagrelor, and very possibly a third blood-thinner depending on your deemed risk. You will almost certainly have a coronary angiogram, usually the next morning, and be monitored closely in the meantime. 

What causes these blockages? Over time, plaques can build up in the coronary arteries like rust on the inside of a steel pipe. Sometimes a piece of these plaques rupture and travel downstream before getting stuck in a narrower section of artery. 

Other than the ECG, we will take bloods looking for further evidence of a heart attack - namely the troponin level. Troponin is a marker that will rise when the heart muscle is under strain from a lack of blood supply. Sometimes, it will take a few hours to see this rise - which is why the doctors will repeat the blood test a few hours later. This also allows us to uncover a possible rising trend. 

Coronary Angiogram 
This is a very important intervention, often life-saving in fact. 
Most commonly, you will be sedated for this procedure before a small catheter (tube) is passed into the artery in your wrist or groin. The cardiologist will pass this along until reaching the aorta (the big vessel near your coronary/heart arteries).
The cardiologist will be able to visualise your coronary arteries by releasing a small dye (contrast), and an X-ray above your chest will be showing them the arteries, illuminated by the dye, in realtime. 
When the blockage is discovered, it will be removed. To prevent further blockages in this area, a stent will often be placed to keep the arteries nice and open in the future. 

Unfortunately, we can't always save the muscle in time. And this can lead to long-term damage affecting the heart's pumping function. It can also lead to a disruption of the heart's wiring, meaning that irregularities in the rhythm can occur.
We will check for these possible consequences with other tests and your cardiologist will commonly advise beginning other medications to counter these issues.