Potatoes are hardy plants that grow in different conditions. There are different varieties as well, and each has slightly different watering requirements. Potatoes generally need less water than many other plants to yield a crop of healthy and sizable vegetables.
Potatoes need one solid watering every week. Accounting for naturally occurring moisture is an important when judging how often to water. A light shower doesn't penetrate deeply enough into the soil, so you should still proceed with your weekly watering after receiving one. Heavier rain might mean shifting your watering schedule a few days ahead.
How you water plants is as important as when you water them. Potatoes have different requirements than some other vegetables. Apply between 1 and 2 inches of water during your weekly watering sessions. Use that amount as a gauge for deciding whether or not to postpone your waterings based on the amount of local rainfall received in a week.
When Watering is Most Important
Potato plants sprout and grow in the early stages quite well even if water is slightly scarcer then normal. The goal when you grow a potato though, is not to create a green, leafy plant, but to get rich and large tubers under the earth. Once a plant has potatoes roughly as big as a marble, it becomes more important to make sure that you stick to your weekly watering schedule if it isn't raining.
Further Watering Tips
Water your potatoes according to a regular schedule, and only shift it as much as needed to account for rain. An irregular watering schedule contributes to cracks and other problems with the potatoes themselves. Observe the soil daily when growing potatoes. The ground should never be muddy and saturated. If it starts cracking, it is too dry and the potatoes need water.