Atompunk is an aesthetic centered around a view of the future from the perspective of the 1950s. It tends to use a distinct, brightly-colored art style. It often depicts imagery associated with "traditionally American" values, particularly a belief in the nuclear family and the suburban lifestyle.
Atompunk envisions a utopian future characterized by bubble domes, glittering cities, and hover-cars for everyone, all powered by nuclear energy.
Atompunk fashion tends to draw heavy inspiration from how people in the 1950's/1960's viewed how the future was going to look, so a lot of the outfits seen in pulp sci-fi of the time tend to be the primarily focus on Atompunk fashion, as well as t-shirts adorned with graphics invoking the covers of pulp sci-fi comic books of the time (which did tend to be rather sexually suggestive from time to time) being part of the Atompunk fashion aesthetic.
Atompunk as a cultural phenomenon originated in 1945 and continued until 1965. It is the period of retrofuturism following and slightly overlapping with World War 2's dieselpunk, emerging after the end of the Second World War. It was also preceded by raypunk or Raygun Gothic, a more specifically aesthetic movement (whereas Dieselpunk and Atompunk are more content-focused). Its optimism was powered by the relief of the Allied victory and by the belief that "if we can split the atom, we can do anything." It encompassed the era of mid-century Modernism, along with much of the Jet Age, the Atomic Age and the beginning of the Space Age. It dropped off somewhat around the beginning of the USA's involvement in the Cold War and gradually faded into pessimism.
Since 1965, most or all instances of this aesthetic have used it with the understanding that it is retrofuturistic (as opposed to simply futuristic, the way it was when it was created). The term "atompunk" most likely originated in 2005, long after the first wave of the aesthetic died down.