Fernet Branca is a digestif amaro invented in 1845 by Bernardino Branca in Milan, Italy. The formula, like many amari, is a closely held secret - but supposedly contains everything from myrrh, chamomile, aloe, bay leaves, to galangal, and zedoary. It spends a year in oak barrels. The "Eagle and Globe" logo dates to 1893.
Fernet is a style of amaro — generally the most bitter and least sweet.
Fernet Branca is dark brown, opaque, strongly menthol-flavored, and complexly herbal smelling. It's also incredibly bitter, with very little balancing sweetness found in other amari. As such, it's fantastic after a large meal, and it's become popular as sort of a shibboleth among professional drinkers. Have a cocktail or two at your favorite bar, then ask your bartender if they'll join you in a shot of Fernet Branca and bask in the good vibes you've just created.
In small doses, Fernet Branca can give a medicinal, spicy backbone to cocktails.