If you insult someone or dismiss them in a hateful way, you're being contemptuous. The difference between being hateful and contemptuous is subtle. It involves disdain.
Being contemptuous of someone or something means that you're combining a deep dislike for them with condescension. To insult someone is a good way of showing contempt, i.e., of being contemptuous. "You disgust me," is one of the simplest and most contemptuous things you can say to someone. "You're stupid," is another. If you can think of something that both hurts a person and suggests that you're looking down on them, you've come up with a contemptuous thing to say.