Most shoes have thick soles, plenty of arch support and little flexibility - whilst this may be important in some endeavours, running is typically not one of them. Such shoes promote heel striking rather than forefoot, which can lead to increased risk of a number of common lower limb injuries including plantar fascitiis, Achilles tendinitis and shin splints.
Minimalist shoes are the opposite of your typical running shoe. Designed to mimic barefoot running, they have thin soles, are lightweight, flexible and offer little arch support. They encourage you to run naturally, landing on the forefoot (balls of your feet) and help develop many supporting muscles of the foot, ankle and leg and improve balance/proprioception.
Examples of such shoes include the Vibram FiveFingers, New Balance Minimus, Nike Free, Merell Trail Gloves, VivoBarefoot Evo and Puma Faas.
Whilst you may not be able to do all of your training straight away in them (it will take time to adjust and a gradual progression is recommended to avoid overuse injuries) all athletes can benefit tremendously by including minimalist shoe and/or barefoot running in their training.
In addition they are also great for weight training - allowing much better transfer of force than traditional highly cushioned running shoes.