"a hair of the dog that bit you" means a small drink of alcohol the day after you drank too much alcohol and are hungover. Some people say that if you are really hungover (sick from alcohol), you can have a small drink of alcohol and that will help your body feel better (lessen the headache and the sick feeling in your stomach). The "hair" is just one small drink. The "dog that bit you" is all of the alcohol you drank the night before.

Hi yoko. It means 'mukaezake' in Japanese. I was surprised when I have hard 'a hair of the dog!' that a way to exprain about the situetion what drinks a lot of alchole and have hungorver.