I was first introduced to this cocktail in the 90's at a seafood restaurant in Santa Monica, California. The name of the restaurant eludes me, perhaps it was too many of these delicious beverages, but nonetheless, it was pitched to me as the "James Bond Martini."
Not being much of a martini guy at the time, I figured what the heck, if it's good enough for Bond, I better try one. It was clean, crisp and refreshing. It now stands as my go to, in place of the traditional vermouth-laced version.
I can tell you about the story of the name and how it came about in the Ian Fleming novel "Casino Royale"; but Wikipedia covers it far more thoroughly here if you must know: https://en.wikipedia.org/wiki/Vesper_(cocktail)
BUT! If you want to see how the double-O agent himself makes it, you can see it right here: