An extremity is a limb or appendage of the body, particularly the hands and feet. If the weather is cold, you have to be sure to protect your extremities from frostbite by wearing gloves, warm socks, and boots.
The noun extremity also means the outermost point or part — the one that's farthest away. Nauset Light Beach on Cape Cod, for example, is the easternmost extremity in the state of Massachusetts. Extremity can also refer to the greatest degree, especially of an emotion or physical feeling. The extremity of the pain in your ankle might make you realize you should go to the emergency room for x-rays.