Flatfootedness is also called pes plano valgus or pronation. The calcaneus (heel bone) angles inward and the arch drops. Without support, the lack of an arch can cause foot pain and fatigue. Injury is a possible result.
When running, the heel of a normal foot hits the ground first. Some of the force is transferred or translated to the medial foot (arch of the foot). The arch flattens as it absorbs some of this shock. But if the foot is already flat, the extra load and force can cause problems over time.
There are several shoe designs that can help with this. The first is a balancing bar around the base of the heel. This bar supports the calcaneus in a neutral position and holds the foot steady. This keeps it from pronating too much.
A second design feature is an arch support. Simply placing the foot in a position that mimics a normal arch can be helpful. Some shoes have a motion control design to help absorb the shock.
Different materials are used in the midsole of the shoes. A soft flare on the outside of the shoe slows the pronation movement of the rear foot. At the same time, a firmer material in the midsole along the medial (inside) of the shoe helps prevent excessive pronation.
If possible, shop at a store where athletic equipment and clothing are sold. Ask for help finding the right kind of shoe for your interests and activities. Unless you have a severe foot deformity, this may be all you need.