The word "meunière" is a reference to the miller of wheat, whose wife (acccording to French lore) cooked everything coated with flour. The original French style of trout meunière, then, was seasoned and floured, sautéd in butter, and then topped with the browned butter from the pan. This is still more or less how the dish is done in some restaurants—notably Galatoire's.
There is a New Orleans twist on this, however, and I think it's better than the French classic. It was invented by "Count" Arnaud, who, while trying to standardize and stabilize the sauce so the fish could be fried instead of sautéd, added a bit of stock and roux to the butter and lemon. At its best, this sauce is incredibly good and works not just on trout but also on other fried seafood, notably oysters.
