To moisten is to make something slightly damp or just barely wet. It used to be necessary to moisten a postage stamp with your tongue before sticking it on an envelope, but today stamps stick without being licked.
A misty rain will moisten your hair, and running uphill moistens your forehead with sweat. Your houseplant that prefers a small amount of water will be happiest if you barely moisten the soil. While rain overnight will leave your front lawn soaking wet, the morning dew will simply moisten each blade of grass. Moisten comes from moist, from the Old French moiste, "damp." The Latin root means something closer to "moldy or slimy."