When you're "addicted" to something, like drugs, it means that you have to have more and more of it, and you can't stop yourself. You call this problem "an addiction". But people also use the word "addiction" to talk about less serious things that they can't get enough of, such as:

I'm addicted to "24", so I don't know what I'm going to do now that it's over.

To "break an addiction" means to stop yourself from being addicted.