There are lots of ways to mess up a given recipe, but only a few ways to finish a tough piece of cooked beef. It's essential to choose the right cut for the cooking method you have in mind. Some cuts are tough and chewy, and need long, slow cooking. Others are tender enough to cook relatively quickly, but usually cost a bit more.
That means toughness often comes down to your choice of cooking method. Quick, hot cooking tends to make muscle tissues contract and toughen, so it's only a good option for the tenderest cuts. Slower cooking methods will make tough cuts tender, but overcooking a tender cut can make it tough and dry. You can work around each of these scenarios; you just have to adapt your technique to your situation.
Tenderizing Roast Beef
Suppose, for example, you've make a beef roast and it turns out be really, really tough. You have a few different options. If you need to get dinner on the table right this minute, your best bet is to slice the beef as thinly as you can get it. This shortens the muscle fibers, and minimizes the effect of any chewy connective tissue, so it works for any cut regardless of whether it's undercooked or overcooked.
If your beef is tough and dry because it's been overcooked, the thin-slicing technique will tenderize it but you'll still need to moisten it. If you have plenty of gravy, that's the obvious solution. If you don't have gravy or pan juices to work with, a bit of homemade or store-bought beef broth will help a lot. You can even cheat and use a "cream of something" soup as a sauce.
If your roast is a really tough cut, like a brisket or chuck roast, it's best suited for cooking as a pot roast. If you have time, fix the problem by simply cooking it longer. Turn down the temperature and put it back into the oven, ideally in a roasting pan or Dutch oven with a bit of water or broth added. Keep cooking until the roast's fat and gristle have broken down, leaving it luscious and fork-tender.
Tenderizing Other Cuts
Variations on those same techniques apply to other cuts, and sometimes are the whole point. With a flank steak or skirt steak, for example, you're not likely to serve it as a plain old slab of beef on a plate. Instead, to tenderize flank steak, you cut it across the grain of its long, stringy muscle fibers. You can still cook it to medium rare, or however you like your beef because slicing it thinly makes it chewable.
Turning down the temperature helps with many cuts, especially with pot roasts or any cuts braised in liquid. Cooking at a boil makes meats shrink and toughen, and once that happens, they'll take a long time to tenderize again. That won't happen with a gentle simmer, so your beef won't toughen as it cooks.
If you're not certain whether a piece of beef is best for quick or slow cooking, look it up. Most recipes specify which cuts work best, and you'll seldom go wrong using the ingredient a recipe asks for. If you're winging it, or can't find a recipe that uses the cut you've got, industry sites like Beef: It's What's for Dinner can tell you which cuts work best for which cooking methods.
Sometimes the conventional "saves" aren't a viable option. If your stew is done but the beef is still tough, you can't just keep simmering it because overcooked beef stew is mushy and unpleasant. Instead, separate the beef and cook it until it's fork-tender in a bit of broth or some thinned gravy from the stew, then add it back. That way the beef will be tender, but the vegetables won't disintegrate.
For really tough pieces of beef, your final option might be to mince it up. Cut away any tough pieces of connective tissue, then chop the beef vigorously with a sharp knife, or buzz it in a food processor until it's broken up. You can use the chopped-up beef in most of the same ways you'd use ground beef: Add it to pasta sauces, use it to make chili or a shepherd's pie, or even drown it in sauce as part of a casserole. Chopping it up makes the beef tender, and you'll still add all the same flavor to your finished dishes.