If you take the oat, the whole oat, and take off the hull but leave the bran on and don't process it any further, you have a groat. When I eat groats I usually toast them in a little butter then simmer for at least 30 minutes. These are good but chewy and take a while to cook. They're good but take a lot of effort, sort of like brown rice. I hear they're good in haggis too, but I'm far too chicken to try one.
Scotch oats, Irish oats, or pinhead oats are what you get when you take a groat, remove the bran, then send it through steel cutters. These are also called steel cut oats. I love these: toast in butter until they smell nutty, then simmer for a while adding salt and cream at the end. What you get if you do this is a pretty awesome hot porridge which I think is pretty magical--tasty, kinda chewy, with a nice creamy texture even if you don't add dairy.
Take the same groats, remove the bran (usually), then mash 'em flat and you have rolled oats or "old fashioned" oats which is typically what we think of when someone says "oats". You can cook them in the microwave but they've lost the awesome; convenient but iffy. If I have to, this is what I buy when I want oats and can't find any steel cut oats or don't have the time to cook them (say, at work). These are great for things like oatmeal cookies and granola which I almost always make myself because it's a) dirt cheap, b) kinda fun, and c)I have a nut allergy.
"Can you process them more? Well, sure!"--Alton Brown
"Quick oats" are those that are cut, par-cooked, then mashed even flatter. They blow away in a stiff wind. They cook in a tiny fraction of the time, like a couple minutes of nukeage. They don't get nutty. They don't get creamy. They don't, in fact, taste all that good at all, these "instant oats". I'm not sure they're terribly good for much of anything beyond being convenient. I hate these and almost never buy them.
I almost never buy them. In an attempt to eat healthier, I bought a big carton of Quaker Oats. Guess which kind I got.