A quick heads-up before diving into the story: I've set aside for you a complete shopping list of every ingredient you'll need so you can recreate this classic at home. Here's the tale.
The Moscow Mule is the product of a very American miracle: three struggling salesmen meeting at the Cock 'n' Bull pub in Los Angeles in 1941. John G. Martin owned the unknown Smirnoff vodka brand, Jack Morgan was trying to push his ginger beer, and a third partner, often named as Sophie Berezinski, was stuck with a container of unsold copper mugs shipped from Russia. Combining their troubles into one drink was pure marketing genius.
Martin famously traveled bar to bar with a Polaroid camera, photographing bartenders holding the copper mug and then showing the picture to the next bar as proof everyone was already serving it. That bluff is credited with launching vodka in the United States and cementing the cocktail's iconic copper vessel, inspiring a whole family of "Mules" built on the same template.
To recreate this one with the right style, tools matter as much as the recipe. I use and recommend a complete barman kit like this set on Amazon, which covers every essential tool at a very reasonable price. For a more serious setup, a professional cocktail bar station keeps bottles, glasses and tools organized and close at hand, instantly upgrading the whole experience.