Neutral Cushioning Shoes
A Neutral Cushioning shoe (Men's | Women's) is best for runners with a high arch who do not pronate effectively. These shoes do not have medial supports but are more concerned with midsole cushioning. The midsole will provide the extra shock absorption that the lack of pronation is missing. Along with a runner who does not overpronate, Neutral Cushioned shoes (Men's | Women's) also work well for midfoot and forefoot strikers.
Motion Control Shoes
Motion Control shoes (Men's | Women's) are for runners who generally have a low or flat arch and are moderate to severe overpronators. These shoes employ extra support devices on the medial side to slow excessive pronation and tend to have wider and flatter outsoles. Heavier runners who need extra support and durability may also want Motion Control shoes (Men's | Women's).
Stability Shoes
Stability shoes (Men's | Women's) are for runners who have normal or medium arches who are mild to moderate overpronators. These shoes have some medial support and good midsole cushioning. Because normal or medium arches are the most common foot type, most runners will need Stability shoes (Men's | Women's).
Trail Shoes
A Trail shoe (Men's | Women's) is for a runner who desires better traction and durability for off-road runs. These shoes tend to be more weather and water resistant and are slightly stiffer for better stability on uneven terrain. Trail shoes (Men's | Women's) can be Stability, Motion Control or Neutral Cushioning depending on the model and are suited for different runners the same way road shoes are.
I hope this helps save money and avoid injuries caused by wrong choice of shoe.