It's the concept that addiction is a disease that affects mind/thought process, emotions/feelings, and our physical bodies.
The concept isn't a new one, but it is just recently becoming more widely accepted among universities, medical professionals, and other authority types- namely those working in the field of addiction. It's an attempt to deepen our understanding of the phenomenon and move away from stigmas associated with drug use, abuse, and addiction. Addiction is more than just bad behavior, lack of will power, or lack of caring. Drugs that are commonly abused affect parts of the brain used to control and reinforce behavior. For many reasons, some still yet to be fully understood, the brain of someone addicted to drugs does not function the same as a person not addicted. Drug use changes how a brain functions, and these changes can "encourage" drug use/addictive behavior as well make it harder to stop. These changes can affect how we feel and how we think, ultimately affecting behavior.
The changes can even lead to physical symptoms, such as muscle spasms, constipation, eating more/less, sleeping more/less, etc. Some of the physical symptoms are a result of the drug interaction in the body, not brain. Alcohol is absorbed into the blood and travels to every organ of the body thus producing a physical effect apart from the buzz/drunkenness/commonly known effects of alcohol.
Addiction is a biopsychosocial disorder, meaning it affects: the biology of the person(physical), the psyche of a person(mental), and the friends/family of the person(social).