Stability
Stability shoes are those that will be best for the average runner. They have a fair amount of flex and cushioning, and allow a normal foot to move as needed while running. The Asics Gel and GT series of shoes have won numerous awards from Runner's World, including Editor's Choice and Best Buy in the stability category. The Brooks Adrenaline GTS series also has a few award winners, and New Balance makes several stability models.
Motion Control
Motion Control shoes are stiffer through the arch and sole and are designed to help the over-pronator or flat-footed runner maintain control through her stride. Before buying a pair of these shoes, you should have a specialist look at your footfall to be sure this is the best design for you. Try every pair on to be sure they really do control your step every time. Saucony, Mizuno and New Balance all produce a few models of motion control shoes that may work for you.
Trail
Trail running shoes are great if you spend most of your time off the beaten path. Designed with a thicker and more rugged sole, these shoes are made to keep your foot safe and your body stable on variable terrain. While they can be worn on paved roads, doing so is a bit of overkill for what you really need. Nearly every brand makes a trail model, but a few brands that usually specialize in heavier-duty hiking footwear also make a trail runner. In addition to the brands you'd usually consider, add Salomon, Vasque and La Sportiva to your list of potential trail running models.
Racing
For serious racers, having a light-weight, responsive shoe is key to their success and racing plan. In this genre the field narrows considerably, as not every brand makes a shoe specifically for racing. Asics leads the way with a few different racing models, but Saucony and Nike also make shoes worth considering.