The idea of atomic bombs was well-established in popular culture by 1945. (I have written a longer comment on this here.) The idea of nuclear fission was known since 1939, and so was the notion that it might be usable in weapons. But whether you could actually build one or not, and what it would be, was not known — and the reason is because that requires a lot of detailed knowledge about what would be actually going on inside the bomb, and that required a lot of research to find out.

To make an analogy, from my earlier comment:

The analogy I like to use is that of the "warp drive" today. We've all heard of such a thing, we all know more or less what it might mean (faster than light travel, Star Trek, etc.). The geeks among us know that there are hypothetical means of maybe making it work (e.g. the Alcubierre drive), but nobody but a crank would think they are just around the corner. Now imagine that President Obama announced tomorrow that a warp drive had been developed and gone around the solar system ten times as a result — you'd be shocked. But you would recognize the term "warp drive," even if you had no idea how they had actually pulled it off. This is approximately what the public understanding of the term "atomic bomb" would be in 1945 — the idea of atomic energy was known, the idea of making a bomb out of it was known, the specifics were not known.

As for which nations knew about it, the United States and United Kingdom were working on it. The Soviets had spies in that program and knew about their work. Other than that, the Germans knew nuclear weapons were in principle possible but had concluded they were difficult to make. Ditto the Japanese. So no nations other than the US, UK, and USSR had realistic notions of what the weapons in World War II would look like before they were used, and in all of those cases the number of people who knew that was relatively small (on the order of a few thousand maximum).

The Soviet Union was in possession of schematics and documents stolen from the Manhattan Project (largely by British physicist Klaus Fuchs). This gave the Soviet Union full knowledge of the bomb's capabilities, and allowed them to create their own 'virtually identical' model 4 years later in 1949. Other allied nations (the UK and Canada) were participants in the project, so their governments were also aware.

As for the Axis, both German and Japan had limited atomic bomb programs (neither got very far, despite allegations that the Japanese successfully tested one). As a result, they would have been able to estimate the destructive capabilities of an atomic bomb, and assumed that the Americans were also working on one.

The Soviet newspaper Pravda published an article in October of 1941 that mentioned the idea of an atomic weapon that could destroy a large city. Nuclear fission was known within the scientific community in the 1930s, and at least to physicists the destructive potential of a nuclear chain reaction was fairly obvious.

The thing that wasn't obvious about the atomic bomb was if it was practical to build one using technology available in the mid 1940s (just barely as it turned out) and if a crash program to produce an atomic bomb could lead to a weapon that could be used before the war was over (again, just barely).