Vermilion is a deep, brilliant shade of red. You could describe your grandfather's reddish-orange scarf as vermilion.
The specific pigment know as vermilion was originally made from mercury sulfide, a chemical compound that produces a deep red color. This kind of vermilion was most famously used in enameled Chinese lacquerware. However, you can use the word vermilion to describe anything that's a particularly deep, rich shade of red. The root word is the French vermeil, "bright red," from the Latin vermiculus, "a little worm," or the insect that was historically used to make red dyes.