What does it mean to braise?
Back when cooking was done around the glow of a warm fireplace, braising meant sticking whatever it was you were going to eat in the embers of the fire and leaving it there until it was done. These days, to braise means to cook in a little liquid, usually over a relatively long period of time.
To braise, you need a pot with a well fitting lid -- think Dutch oven or tightly covered sauté pan, something that can work both on the stovetop and in the oven. A heavy pan (like enameled cast-iron or stainless steel) is good, too, since it evenly distributes the heat through the food and is less likely to scorch on the bottom.
Braised meats are often seared first in oil or butter. Then, aromatics, spices, vegetables and some cooking liquid -- wine, broth, water -- are often added. The pot can either stay on the stovetop or go into the oven. Leaner meats like pork chops and chicken breasts usually braise for less time on the stovetop. Fattier, more flavorful cuts go into the oven and cook longer.
Braised vegetables are usually started off with a quick sauté in butter or olive oil. The liquid is added, then the pot is usually covered and the vegetable cooked on the stovetop for however long. Any vegetable can be braised, but greens are an usually good choice.