The Rotten Apple Award is a trophy of sorts shaped like a rotting apple, as the name implies. In The Gift of Caring, No Heart tells Shreeky she must do something very nasty and uncaring to earn it as a prize.
Shreeky fails to earn it because of the Care Bears being happy, and it is eaten by Beastly to satisfy his hunger he had during the entire episode. He turned green from eating it.
It is unknown if the award is a metal statue or an actual rotten apple, as which it surely would stink, as Beastly eats it.