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While it may seem contrary to usual meat-buying logic, tougher or less-tender cuts of meat are perfect for braising. Pot roast, chuck roast or any inexpensive cut may be used. The muscle fibers and connective tissue that make these cuts stringy or tough are broken down by braising, which gelatinizes the collagen into a more palatable texture. Low heat and long cooking times are used to make a tough cut of any kind of meat moist, tender, and delicious, when cooked properly. Common cuts of beef for braising include:  top blade roast eye roast seven bone, or center-cut pot roast shank ribs or short ribs brisket It's unlikely that you would ever want to braise any kind of lean steaks or loin. You could, but because these meats are tender already, it'd be a bit of a waste. Other than a pot and your cut of beef, the only other absolutely essential ingredient is a liquid in which to simmer the meat. Because this is an opportunity to add a dash of flavor to the dish, it's most common to use wines, stocks, or other flavorful liquid, as opposed to water. Common braising liquids include:   Beef stock or broth. You can match the stock to the dish by using a beef-based broth or stock, although using chicken stock would is universal for braising any type of meat, and might add a nice complexity to your braised beef. Stock is just broth that hasn't been seasoned, so stock is generally better for braising, since it allows you to control the salt-levels, but either is acceptable. Just go easy on the salt if you use broth.   Red wine. Dry red wine can add a nice acidic note to beef, especially when combined with another braising liquid, like stock. The alcohol cooks off, resulting in a rich and fragrant dark sauce. Especially fruity or sweet red wines would be less desirable, but fine if paired with an equal amount of stock to cut the sweetness. The fruitiness of a white wine would pair better with chicken or pork. Since it'll season your dish, make sure it's something you'd like to drink--pour yourself a glass for "research."  Dark beer. English cooking at its finest. Stouts, porters, or black lagers all lend a rich sweetness to beef, and a malty depth of flavor.  The darker the better, when it comes to beef. Some Belgian ales might also work nicely, but experiment and find a beer you with a nice flavor. In general, lighter pilsners and lagers are more appropriate for chicken or pork. How much liquid you'll need will depend on the amount of meat you're braising, and the addition of extra vegetables. As a rule of thumb, you'll want enough liquid to cover the vegetables at the bottom of the pot and come up just to the level of the meat. You're not boiling the meat, or stewing it, so you won't submerge it in the braising liquid. It won't take much, and you can always add additional water to the pot if you didn't have enough wine left in the bottle. Sounds fancy, but it's not. In French cuisine, braised beef and many other meat dishes will always start with a vegetable base of finely minced carrot, onion, and celery, called mirepoix, which is used to pair with the meat and enrich the sauce. After searing the meat, the mirepoix is added and browned briefly before adding the braising liquid to the pot.  For a proper braising, there needs to be something at the bottom of the pot other than the liquid, to give the sauce fragrance, substance, and character, as well as to keep it from drying out. When minced very small, the mirepoix will mostly disintegrate into the liquid over the course of the long cooking time, to flavor the sauce, though you could leave larger chunks to do more of a "pot roast" style braised beef. Depending on the cut of meat, you might use around 2-3 carrots, 2-3 celery stalks, and a small white onion. Depending on what you want to do with the beef you braise, you might elect to make a one-pot meal with the addition of vegetables. In most braising, some variety of aromatic vegetable will always be used to keep the moisture consistent in the bottom of the pot, as well as to release other flavors and aromas. Cooking beef low and slow is a great opportunity to cook vegetables as well.  Other vegetables like potatoes, cabbage, peas, mushrooms, greens, leeks, or other root vegetables can be added to the pot later, about 45 minutes before the meat is done cooking. Some fruits, like apple or pear, can also pair nicely with braised beef, depending on the season. Use firm, slightly under-ripe fruit, if you want to experiment.  Aromatic herbs like rosemary, sage, bay leaf, or thyme can kick your braised beef up several notches. If you've got access to an herb garden, or just want to buy some fresh herbs from the store, tie up a bundle of a few stalks in twine and add at the same time you add the braising liquid. Braising starts on the stove and moves into the oven, making it important that you start in a pot that's oven-safe. Enameled cast-iron pots are perfect for braising, featuring the heat-retention of cast iron and the hefty weight of a good baking dish.  Skillets aren't generally big enough to hold all the braising liquid, meat, and vegetables required for a good braising, while thinner sauce-pans won't hold the heat as effectively as cast iron. If you don't have a cast iron dutch oven, though, anything you can cover and put in the oven will do in a pinch. If you don't have an oven-safe stew pot, but have a heavy-bottomed saucepan, it's perfectly fine to braise meat on the stove-top as well. Some cooks prefer the oven method because it more evenly-distributes heat throughout the meat, while others prefer the simplicity of braising on the stove. Both methods result in tender and delicious beef.

Summary:
Select an inexpensive cut of meat to braise. Select a braising liquid. Start with a mirepoix or some mix of finely minced vegetables. Choose additional vegetables to add as well. Always use a heavy-bottom stew pot or dutch oven.