Basically speaking, it's someone who's a 'queen' at being 'dramatic.' To be considered 'dramatic,' you have to overeact, usually by exaggerating your gestures, your facial expressions, and your body language so that a person in the back of the audience can get your meaning. It's usually used to get attention.
It usually is a term for females. It has only seen it applied to males when it's a gay person in mind.
Being a drama queen is a negative term that means your "over doing it" when you react to something, for example:
1) Stubbing your toe, then screaming and crying and yelling and throwing stuff
2) Being picked on by friends, then screaming and crying and yelling and throwing stuff
3) Being dumped by someone, then screaming and crying and.... you get the picture.