A physical addiction is where your body adapts to the new substance, like heroine and alcohol, basically creating a dependance on it. Because of that dependance, quitting becomes very hard. The body expects the substance, and will actively crave it just like it would for food and water.
A mental addiction is where the mind is used to the substance, often manifesting as a "why not" attitude towards the substance, which can easily turn to a dependance. Ecstasy and marijuana have this effect, but so do things like gambling. Some consider this an "easier" addiction since going cold turkey doesn't produce physical withdrawal symptoms. The mind is much harder to outsmart than the body though, so it's really just out of the frying pan and into the freezer.