There are many factors, but to generalize:
1,) The right meat for the correct cooking application. If you want to make a pot roast, or anything using a slow-cooking procedure, a cut with plenty of connective tissue is a MUST. That tissue, under low heat, with moisture (braising for instance) becomes finger-licking good gelatin... see ribs.
2.) Marbling of fat: This is a matter of what you like in your steak, but also informs you how to cook the meat. A piece of tenderloin should be treated carefully, because as a lean cut it's going to be prone to drying out. A rib-eye or porterhouse can be cooked for a longer period to allow the marbled fats to do their magic.
3.) How you cut it: THIS MATTERS. Take a nice skirt steak for instance; this can be perfect for a nice marinade, or a thin-sliced pan application.
The biggest factor is the balance between the amount of connective tissues you need to reduce to gelatin, intramuscular fat, and lean protein. Once you grasp that, any cut of meat can be cooked to perfection.
Cyrus: The "shoeleather" effect occurs when meat with plenty of connective tissue is cooked for a short time, or at HIGH heat. The actin, collagen, and other goodies contract and turn into little bits of rubbery death. Use moisture and medium-low heat over time and you get gelatin. Ribs are the best example: They have TONS of connective tissue, but when cooked for many hours they literally fall off the bone. Much of the unctuous feel is from gelatin, not fat as well. In a stew, the same happens, and the gelatin goes into thickening the stew and giving it texture.
When you make a stock, the reason you use carcass and roasted bones is for the benefit of flavor, but also to render the rick connective tissue to gelatin. If you clarify the stock with a protein and acid mixture, you get consomme, which in the fridge will turn into meat-jello.
So, if you're going to slap a steak on the grill, a medium cut of sirloin, a porterhouse, T-bone. delmonico, or other cut that is relatively far from the head and hoof is a good choice. If you're making a stew, you can use pretty much anything, but a brisket requires a tough cut and a big one... hence the use of brisket. Stew meat is cut, so it can come from a variety of sources.
Finally, as Ivan Seeking said, there is the matter of age of the cow or bull, but that's not something you can discern from sight AFAIK. I know that in the case of chickens, you once again choose a younger chicken (roaster-broiler) for most applications, but an older hen, also called a Stewing Hen, is great for braising, poaching, and... yeah, stew.
As for dry aging, you can do that at home with a really good piece of beef. If you have a nice roast, it's going to concentrate the flavor of the beef, and some other chemical goodness will occur. Some people love the taste, and others loathe it.
