A proprioceptive exercise is anything that is done off balance in order to develop strength in stabilizer muscles. To challenge your balance you can stand on one leg, or use tools like a BOSU or Airex Pad, like the one in the picture above to decrease stability.
Proprioceptive exercise is often used in rehabilitation. This type of exercise decreases chances of joint injuries, or reoccurring injuries.
Strength coaches and trainers use this type of exercise in elderly people who are at high risk of falling if they get off balance. They can also do this with athletes who depend on balance in order to perform well in their sport.