Organic means physical disease such as a type of dementia and non-organic means non-physical presence of disease usually a psychiatric disorder. So for example if someone has become confused and delirious both causes need to be considered organic and non-organic. Organic causes can be as simple as a urinary tract infection or more concerning such as new onset dementia (often these both can be present at the same time) if no organic cause can be found then non-organic causes are considered, such as major depression etc.
