Why do I feel a sharp pain in the middle of the bottom of my foot when I run?

Possibly nerve. The sharp pain sensation can be the result of a nerve that is being irritated in the arch area and/or the result of a ligament/tendon that may be pulled while you bear weight. This is something you want to see a foot specialist for in order to have your foot checked, gait evaluated and shoes checked. Don't run through it, not checking it could eventually slow you down.
Pain is not good. It is not normal to feel this. Shoes could cause this. Or you could have pain from muscles, ligaments, tendons, or joints. Have your local podiatrist check this out for you so you can run more comfortably.