This is someone who influences or controls a public figure behind-the-scenes.
In 17th century France, Cardinal Richelieu was known as “The Red Eminence,” because of his position in the church and his red robes. However, many thought he was heavily influenced by an assistant who who different colored robes. That person was referred to as “The Gray Eminence.”
“Éminence Grise” is a direct French translation of “gray eminence.”
I feel like the “gray” adjective also describes someone operating “from the shadows,” but perhaps that’s just my personal perspective.
Sometimes also called “the power behind the throne.”
Why I Looked It Up
In the book Too Big To Fail:
As head of the financial institutions group, [Ken] Wilson was Goldman’s top adviser to other banks and respected as an éminence grise throughout the industry.
Interestingly, the phrase is not italicized in the original text, which is common for foreign language phrases.
(Note: to get the correct accent over that “e” on an American English keyboard, I had to type ALT+0233. The capital “E” in the title was just copied from another page.)