The valve is connected to the handle you turn. That handle is connected to the valve stem. At the end of the stem is (typically) a rubber (probably synthetic) washer - a round dognut with a screw holding it to the stem. The stem is threaded so it pushes that rubber washer onto a seat. If the seat is bad or the washer is bad, water will leak. There are other types of valves, but from what you described, it is probably one of those. My guess is that the washer finally gave up the ghost and needs to be replaced. On an old valve, the part it presses onto may have become all rough. This can make the washer degrade quickly, and can prevent it from making a seal. Some of them can be removed and replaced, some can't. Even if it normally could be replaced, if it is that old, it may not come out without a fight. If you are lucky, all it will take is a new washer.