Atompunk, also known as Raygun Gothic, is one of the groups of science fiction Setting Aesthetics based on historical periods - specifically, this is the fantastical sci-fi vision of the 1950s and 1960s, aesthetically modeled after Art Deco, Streamline Moderne, and/or Populuxe (aka Googie). Everything is slick and streamlined, with geometric shapes and clean parallel lines constructed of shiny metal and glass, lit prominently by neon. Sweeping curves, parabolas, and acute angles are used to suggest movement — movement into The Future.
And of course, futuristic fancy-pants technology of the future is ubiquitous. Ray Guns, jet packs, flying cars, Video Phones, Space Clothes, atomic-powered everything, cigar-shaped Retro Rockets and other Shiny-Looking Spaceships, and "electronic brains" capable of calculating complex equations in mere minutes, all decorated with little blinking lights that don't really serve any purpose (but they sure look futuristic!).
Technically speaking, Raygun Gothic refers to the above, whilst Atompunk adds a dash of cynicism and more overt reliance on nuclear power to achieve just about anything, but the terms are used pretty much indistinguishably.
Fallout is basically what you get when you mix post-apocalyptic themes with Atompunk aesthetics.