Antoine Peychaud, a Creole immigrant turned pharmacist, not only invented Peychaud's bitters, but also created an extremely popular cocktail that used them. His 19th century New Orleans pharmacy was often the site of late-night parties, where Peychaud served absinthe and other concoctions as a bit of preventitive medicine for his guests. His signature cocktail featured absinthe, brandy, and a dash of his trademark bitters.
Soon, other local bars (known then as coffee houses) started serving the drink, most notably the Sazerac Coffee House in Exchange Alley, where the drink finally got a name that stuck. When absinthe was eventually outlawed, anisettes like Pernod were commonly substituted, but Herbsaint is still preferred. Rye whiskey like Old Overholt was substituted for brandy so often that it is now the default.
If you are lucky enough to be in a bar where the bartender knows how to mix a Sazerac, it will hopefully be made like this:
Chill a 3.5oz bar glass and set aside. Muddle a cube of sugar in a different glass with a dash of Peychaud's bitters and just a couple drops of water to aid in dissolving the sugar. Pour a jigger of Old Overholt rye whiskey into the sugar glass, toss in some ice, and and mix. Dash several drops of Herbsaint in the empty chilled glass and twirl around so the entire glass is coated. Shake the excess Herbsaint out of the glass. Strain the contents of the rye glass into the Herbsaint glass and twist a lemon over the drink, but do not place the twist inside the drink.