Hmm let's see, I'd generally break down the usable categories of offal as I write below. Generally offal refers to anything beyond normal muscular cuts, so many things considered offals are muscles and not organs.
Beef / Veal / Pig
Blood - Wouldn't want to forget the most metal of all of them. Not usually eaten on its own, usually put into forcemeats like morcilla or boudin noir, or used to thicken stews. Blood has a thickening property kind of like eggs in that it has to be added at the end, and if overheated after adding will coagulate. A traditional civet is made with a game animal, usually rabbit, braised with mushrooms and some root veg in a rich stock and then thickened with blood of the animal. Blood sausages come in all different formulas, boudin noir is loose and pudding-y, while an aged morcilla is dry and hard. Many blood sausages use rice or some other starch to bind all the blood together. The best blood sausages I've had were ones we made at work, with about 40-50% blood and the rest ground pork and fatback.
Brains - Haven't worked with these yet, but have seen them for sale at the Han au reum. From what i gather are treated very similarly to sweetbreads. VERY perishable, you need to get them usually direct from your hog farmer or whoever. Notable for having a ridiculous amount of cholesterol (as much as 30x that of normal red meat).
Bull Penis - had this once at an Izakaya in the East Village. No I'm not making this up. Was chewy and tasteless. Looked like they just boiled the crap out of it.
Cheeks / Bath Chaps / Jowls - from pigs. Usually cured in the same manner as bacon or pancetta. Very fatty.
Chitterlings - hog intestines. Keep in mind these are literally full of pig shit so they need to be very well cleaned and blanched. I haven't yet had these at a restaurant and been glad I ordered them. Maybe I just haven't been to the right place...
Fries (Testicles) - supposedly popular in Texas. I haven't ever worked with them or had them, not for lack of desire but simple haven't run across them.
Heart - hearts are generally well used muscles (logical when you think about it) so tend to be pretty tough, usually trimmed of fat and sinew and braised or stewed, can also be cooked sous vide and sliced thin and seared. Smaller ones from veal or lamb are more tender and can be stuffed with forcemeat and roasted.
Kidneys - usually only from veal or lambs, otherwise they are inedibly gross. Can be simply sauteed or roasted, sliced and served with simple veg or salad.
Lungs (Lights), Spleen (Melt) - from what I understand, pretty much gross and aren't eaten by people who have much of a choice. But hey, maybe I'm missing out. Never had em. Henderson insists they're "the perfect organ to give offal a good name" so maybe I'm wrong.
Marrow - pretty simple, usually bones cut across the grain and then roasted, the marrow fat is taken out and spread on toast. Or marrow can be popped out cold and diced, and used as a cooking fat. Be sure to soak marrowbones in cold water overnight or longer to get as much of the blood out as possible.
Oxtail - Deliciously fatty and well marbles, totally underrated. Usually seared and braised in a brown stock, after which there are many many uses. Traditional Spanish rabo de toro comes to mind: braised in brown stock with sherry and served with chickpeas IIRC.
Sweetbreads - Sweetbreads are awesome. They are the thymus glands, usually of lamb or veal. These need to be stored in milk, and then blanched several times in buttermilk or acidulated water. Then usually trimmed and portioned, sometimes made into a torchon. Usually breaded and fried or floured / breaded and sauteed.
Tongue - Tough cut, usually braised low and slow for a long time in a simple broth or court bouillon and then trimmed, peeled and sliced thin on the slicer, for garde manger type things or sandwiches.
Tripe (Honeycomb primarily, there are other parts of the stomach but this part is most desirable) - the way I've been taught to prepare tripe is to blanch it at least 4 or 5 times from cold water, before trimming and dicing and then braising it. After all that blanching of course it doesn't have much flavor, its mostly a texture thing. This stuff is just packed with collagen that breaks down to gelatin in the braise yielding whatever broth you used with a ton of body. Usually best spicy, a la menudo or with kochu karu or something.
Trotters - usually braise, then meat is picked out and used in a torchon or something, or can be stuffed. Haven't worked with these yet.
Poultry (Chickens and Ducks)
Gizzards - Confited also, after that usually into sausages or forcemeats, haven't used them on their own yet.
Hearts - We confit them and then saute, usually coated in wondra.
Livers / Foie Gras - Livers are usually made into pates or forcemeats of many kinds, and well, foie gras... we could have a whole thread on that alone. Duck and chicken livers are also great simply sauteed, a little salt and pepper, maybe some sherry or madeira to deglaze. When I was a kid we used to saute these up and eat them with a little cheese and toast while we were waiting for the bird to roast.
Necks - kind of tough, best gently stewed or braised, then meat can be picked and used for whatever, or you can leave whole, and sear the neck hard on both sides and serve that way.
Tongues - Very annoying. After confiting them, you have to individually peel each one. And after all that, still not sure there's much to do with them. We got a sample from Hudson Valley Foie Gras and screwed around but never did anything productive with them.
For more information of how to deliciously cook offals, consult Fergus Henderson's books, The Whole Beast and Beyond Nose to Tail. He runs a restaurant in London, St John, that specializes in offal cookery. He is the source of one of my favorite quotes: "If you're going to go through the trouble of killing something, it seems only polite to use the whole thing"
To buy offals, you'll need to find a butcher that specializes in that type of thing. Check out whatever markets or butchers sell to Asians or Latin Americans, those guy get down with the offal. Han au reum usually has lots, including weirder things like pig uterus, but I wonder at the freshness of it all. For most of these things I admit the prep is too extensive for most to bother with at home, all of the above information is from my experience in professional kitchens.