Basically a selfish chore or task is one that only gives points to the person completing it. It will rise the selfish meter at the bottom of the screen and if done too much, will force the family to complete a task altogether while losing points.
There will be a devil sword that symbols this chore. All you have to do is have someone do this selfish chore and have another person stop that player from completing the chore. The stopping part is really simple, you have to tap or click the screen until the bar fills. Do this five times in a game and the achievement is yours.