Bring the water to a boil over high heat, then reduce the heat to medium and let the potatoes simmer until they are fork-tender about 15-20 minutes. Be careful not to overcook them; they should be tender but still hold their shape.
Once the potatoes are cooked, drain them in a colander and let them cool for a few minutes until they are cool enough to handle. Cut the cooled potatoes in half (or quarters if they are larger) and place them in a large mixing bowl.
The Pesto Potato Salad can be served warm or chilled. If not serving immediately, cover and refrigerate. Before serving, you might want to toss the salad again as the pesto can settle at the bottom.