Braising is a form of moist-heat cooking in which the item to be cooked is partially covered with liquid and then simmered slowly at a low temperature.
Braising relies on heat, time, and moisture to break down the tough connective tissue collagen in meat, making it an ideal way to cook tougher or older animals cuts. It also works well with chicken. The best cuts of chicken are the legs and thighs although lots of people like to raise a whole chicken. Use chicken on the bone with skin so you get all of connective tissue. There's really no reason to braise boneless, skinless chicken breasts. You are better off sauteing or grill them.
Braised meats are often seared first in oil or butter which helps to develop flavors as well as making the meat more appealing visually. If the food will not produce enough liquid of its own, a small amount of cooking liquid (water, tomatoes, beer, wine) is added. The pot can either stay on the stovetop or go into the oven. Leaner meats like chicken breasts usually braise for less time on the stovetop. The dish is cooked covered at a very low simmer until the meat is fork tender. Braising causes the muscle fibers to absorb moisture from the cooking liquid and steam which gives you a juicy piece of meat.