What is the damage being an alcoholic does to your organs?

Bad for many organs. Alcohol is a chemical toxin that directly damages the cells of most organs. Most notably, cell loss in the brain causes memory, vision, and coordination problems. The heart can enlarge and fail to pump properly. The liver can fail to clear toxins, remove cellular waste, and make enough proteins to help blood clot. The gastrointestinal system can fail to absorb nutrients, causing malnourishment.
No short answer. Alcohol damages virtually every human organ system - mostly the brain and nervous system, liver, gastrointestinal tract, pancreas, heart and can cause cancer. In social amounts (less than 2 drinks per day), damage is uncommon, and such amounts may be protective against hardening of the arteries. In larger amounts, damage may be severe, and include dangerous behaviors (driving etc.) & depression.