My eight-and-three-quarters year old son would like me to ask what would happen if a person detonated an atomic bomb on the moon.
He asks questions like this all the time that I have no answer for, so help me out.
Probably something along the lines of a small, bright flash and a new crater? I dunno. Only one way to find out for sure, though. Somebody get me a nuke and an Apollo rocket. For science! 
During the cold war, the US Military really, really wanted to do this. They were finally persuaded to not to by scientists who said that a) it had no military value what so ever. b) it would be hugely expensive and c) it would forever alter the pristine lunar environment, making future scientific missions less accurate.
An atomic “explosion” is really a huge release of energy in the form of gamma radiation. Much of this radiation simply passes through the bomb casing and radiates away into space, invisibly. But, some of it is absorbed by the bomb casing and components as well as the lunar surface the bomb is resting on, heating them far, far past their boiling points; this turns them into extremely hot gas which is so hot it glows brilliant blue-white. So, the first thing you’d see is a brilliant flash of extremely bright light. This secondary burst of light in turn heats the ground around and under the bomb, vaporizing it and creating more hot gas and light and this process spread out in a hemisphere around the bomb, cooling as it goes, ultimately resulting in a crater lined with glassy material. The superhot gas also cools as it expands, creating a shockwave that would probably blow up dust from the surrounding lunar surface in a toroidal cloud that would quickly settle out in the airless environment of the Moon. Since there’s no air, there would not be the classic mushroom cloud associated with Earthbound nuclear detonations.
Without an atmosphere, there’d be no shockwave, or at least not a significant one. There also wouldn’t be a mushroom cloud. Yeah, flash and a crater sounds right. Aside from the radiation, it would probably look much like a large meteor strike.
I expect if the blast is big enough, astronomers might be able to detect seismic ripples for a while, rebounding around the regolith.
It’d easily be visible from Earth - as was determined by a bizarre 1950’s Air Force project that investigated what the effects would be, as part of a feasibility study to see whether they could show the world that, well, that they could explode a nuke on the Moon. Carl Sagan was one of the scientists involved.
Even a two to three inch meteor striking the moon is easily visible from earth, using a small telescope:
Leonids Strike the Moon (story, map, and movie)