The type of shoe you require is based on your running gait (style).
Neutral
A neutral runner typically has a high/medium arches and an smooth running style. A neutral cushioning shoe provides even, balanced cushioning through the gait cycle.
Over Pronator
Overprontator's tend to have lower/flatter arches and would roll inwards when running. To compensate for this an over pronator would wear a support shoe which has additional medial (inside) support to correct the overprontation and stabalise the foot. Most runners over pronate and it is a common trait. If you overprontate excessivley or have very flat feet you will require a motion control shoe.
Under Pronator (Supinator)
Supination (or under pronating) is when the foot rolls outwards. It is typically found in runners with very high arches. A supinator should wear a well cushioned neutral running shoe.
How to test what type of shoe you need
The 'wet foot test' is a quick way to test what type of shoe you need.
If you look at your wet footprints on a tiled floor you'll be able to see a clear indication of your foot shape: