When another person's bad luck secretly makes you feel good, that's Schadenfreude. Your brother's rejection from a college that also rejected you might give you a twinge of Schadenfreude.
If you're fired from your difficult job, no one can blame you for a bit of Schadenfreude as you watch your replacement struggle with your old tasks. This German word perfectly captures that satisfied feeling everyone gets at times when someone else runs into misfortune. In German, Schadenfreude literally means "damage-joy," and it's always spelled with a capital S. The word came into English use in the 1920's, and you can spell it with a small s if you prefer.