"mental addiction," or psychological addiction as it's most commonly called, is when you're inclined to crave a substance because you "like the effects" and whatnot, simply put. marijuana can be psychologically addicting for many people.
physical addiction, on the other hand, is coupled with actual physical withdrawl "sickness" (read: symptoms) when the drug isn't present in your body; when your body needs the drug in order to preform functionally, basically. cigarettes and heroin are an example.

something along those lines.