An éminence grise (French for “grey eminence“) is a powerful decision-maker or advisor who operates “behind the scenes” or in a non-public or unofficial capacity.
This phrase originally referred to François Leclerc du Tremblay, the right-hand man of Cardinal Richelieu. Leclerc was a Capuchin friar who was renowned for his beige robe attire (as beige was termed “grey” in that era.) The title “His Eminence” is used to address or refer to a Cardinal in the Roman Catholic Church. Although Leclerc never achieved the rank of Cardinal, those around him addressed him as such in deference to the considerable influence this “grey” friar held over “His Eminence the Cardinal”. 
An éminence grise can alternatively refer to an elderly (“grey-haired”) personage who is renowned for past accomplishments, and now acts as an advisor rather than a principal actor. He might be politically influential as a consequence of his honored status within an influential group or society as a whole. For example, a distinguished retired physics professor emeritus who advises scientific leaders and government officials on nuclear energy; or a retired U.S. Senator who advises the President on an informal basis, etc.