Any runner who overpronates will likely want a stability running shoe. If your foot rolls inward on each step or you have excess wear on the inside edge of your shoes, a stability shoe could be helpful.
While the idea that not wearing a stability shoe will cause injuries has been debunked (and don’t let a pushy salesman tell you otherwise!) - most runners who overpronate will find a stability shoe to be much more comfortable than neutral shoes.
If you’re only a mild overpronator, you could get away with wearing a neutral shoe. The key is finding a comfortable shoe that supports your feet enough to prevent the rolling motion.