Damehood is the rank of being a Dame: the women's title in the British honours system that sits beside a knighthood. A woman given that honour is then called Dame before her name. It is a rare, formal word; you will see Dame in the news far more often than damehood. Though actually, in older or American talk dame can just mean a woman, sometimes rudely — damehood has nothing to do with that slang. It only means the honour, the same way knighthood means the matching rank for a man.