In politics, it most commonly refers to a nominal subordinate or advisor to an officeholder (often called a "figurehead") who serves as de facto leader, setting policy through influence, manipulation, or both.
In recent times, family members and official or unofficial advisers might take on a similar role.
The term typically has a negative slant, implying that the power behind the throne exercises their influence illegitimately, or at least extralegally.
This phrase originally referred to Cardinal Richelieu's right-hand man, François Leclerc du Tremblay (also known as the Père Joseph), a Capuchin friar who wore grey robes.
Because the Cardinal de Richelieu, the power behind the throne of King Louis XIII of France, as a Catholic cardinal was styled Son Eminence ("His Eminence"), his alter ego Père Joseph was called l'éminence grise (which is also the English title of his biography by Aldous Huxley).