In French cuisine, pepper sauce or sauce au poivre, is made of a cooked mirepoix thickened with flour and moistened with wine and a little vinegar, then heavily seasoned with black pepper. More traditional versions in French haute cuisine use espagnole sauce to thicken. In this sense, peppercorn sauce is a synonym.
Across the Americas, pepper sauce often refers to a sauce of chili pepper in vinegar, often accented with garlic, herbs, or other spices. In this sense, it is a synonym of hot sauce.