To mollify is to calm someone down, talk them off the ledge, make amends, maybe even apologize.
Mollify comes from the Latin mollificare to "make soft," and that's still at the heart the word. When you mollify someone, you smooth things over, even if you're maybe still a little mad: "I was angry that the guy took my seat, but I was mollified when he offered me one closer to the band." Unlike the sharp sounds of antagonize, there are only soft sounds in this word that means to make someone feel soft and cuddly. Although dryer sheets might soften your clothes, they don't mollify them (unless your clothes were really mad at you before).