Soaking cut potatoes overnight helps to remove excess starch from them, which makes them less sticky in certain recipes, such as mashed potatoes. Too much starch makes the bits of potatoes sticky, forming what is almost a glue-like consistency.
Cooking the potatoes, then draining them from the cooking water and rinsing them under cool running water also removes excess starch.
In potatoes, there is a starch molecule called amylose, which is what causes the glue-like consistency sometimes found in mashed potatoes. It is important to cut the potatoes into small enough pieces (about 1 inch on each side) so as to expose enough of the potato to the water.