You can abuse a drug without having an addiction. For example, just because Sara smoked weed a few times doesn't mean that she has an addiction, but it does mean that she's abusing a drug - and that could lead to an addiction. Addiction means having no control over whether to use a drug. A person who's addicted to cocaine has grown so used to the drug that he has to have it. Addiction can be physical, psychological, or both.
Physical addiction is when a person's body actually becomes dependent on a drug.
It also means that a person builds tolerance to a drug, which means he needs a larger dose of that drug to get the same effects. When a person who is physically addicted stops using drugs, he may experience withdrawal symptoms. Withdrawal can be like having the flu - common symptoms are diarrhea, shaking, and generally feeling awful. Psychological addiction may happen along with physical addiction or on its own. In this case, the cravings for a drug are psychological, or mental.