So-called Natural Running shoes are unique in the way that they provide a barefoot feeling while you run. This means that the shoes have a very small offset and a thin midsole and, consequently, impact is distributed all over the foot. These highly flexible shoes make you feel like you are running barefoot. Important properties of Natural Running shoes are their low weight and high degree of breathability. Shoes in this category are good to swap with neutral shoes to vary your workouts and put your foot's muscles under varying pressures. However, shoes in this category should not be used for all training runs as the strain on the joints could be too high, especially in inexperienced runners.
A natural running shoe would be, for example, the Nike Free RN Distance.