"Laughing at" someone suggests a mean, hurtful attitude. You're enjoying someone's embarassment or making fun of their weaknesses. Of course, you can "laugh at" a joke or "laugh at" a funny movie without it being mean, but "laughing at" a person isn't nice.

When you're laughing because of something that a person did or said, but you don't intend to be mean, you can say "I'm laughing with you, not at you." "Laughing with" someone suggests that you're both laughing and having fun.