There is a difference between angina and heart attack. In angina, you get narrowing/hardening of the blood vessels of the heart. Sometimes the blockages are fat deposits - or sometimes the blood vessels are sticky and pick up waste materials, fibres etc that make them narrow. When it is hard for blood to get through, it causes pain.
With a heart attack, these narrowed vessels actually become fully blocked, and this leads to death of the heart tissue itself. This is called a myocardial infarction (heart attack). There are a few ways you can tell the difference:
ECG changes will show whether there are electrical changes in the way the heart operates - if there are changes that is one indicator of heart attack having happened.
Cardiac enzyme changes - This measures a few types of enzymes (CK, CK-MB, Troponin I). Each enzyme change will show if there has been any breakdown in muscle at all, and some of them will be more specific to cardiac muscle. So if there are enzyme changes, this is another reason to suspect a heart attack.
A history of heart disease too - basically if you have two out of the three you have probably had a heart attack.
With angina - because the vessels are only partially blocked, the vessels can be dilated (made larger) temporarily using a drug called GTN (aka nitrogycerine or anginine). If the problem is a partial blockage (angina) then the pain will often go away after this drug is given. Your Dad will likely come home with little tablets to put under his tongue or a patch that is applied to the skin.
Angiography is just a way to check which vessels are causing the problem. There is also something called PTCA (percutaneous transluminal coronary angioplasty) where instead of just looking at the vessels, they can insert a small balloon, and even place a stent in there to clear out and widen the site of the blockage.
Chest X-ray can tell him things like if his heart is too big - if he's had heart trouble for some time, then his heart will have been working very hard and some of the muscle will have become loose and floppy. This makes your heart look bigger. It can also find out if there is any excess fluid around the heart, or any infections to the linings surrounding the heart. They will also check the lungs and see if there is anything wrong there, which could account for the breathlessness he is probably expriencing with the angina.
Long term, your Dad can do much to manage his angina. He may have high blood pressure - they can give him drugs to control that (ACE inhibitors, beta blockers, calcium channel blockers etc). They will want to thin his blood so it moves more easily through his vessels - so using some kind of antiplatelet (like aspirin) or anticoagulant (like warfarin) will help. He will want to be able to control angina attacks rapidly, so he will have GTN in patches, tablets or spray. He may also have some pain, so he will proabably be given some pain control.
The kind of angina he has will be diagnosed by his MO hopefully. There are many different types (Unstable, stable, prinzmetals, intractable and silent ischemia)...and they have many different triggers and symptoms. Treatment will be based a bit on what kind he has.