I've always loved humor stories, and relish the idea of making one. But the thing I cannot grasp is the key elements of one. Are the key elements the same as a normal story (Problem, Solution, etc.)? Or are they different?
Same elements, but funnier.
A humor story will have the exact sames elements as a normal story. You will still have a plot, characters, and scenes. You will still need to develop your plot and have character arcs. The largest difference is that the rule of funny applies, which gives you more leeway for events than a serious book.
The other big consideration when writing comedy is pacing. Most comedies have faster pacing than their serious counterparts because jokes that miss are boring and can kill the pacing of a story. You either need to be sure the joke hits or have the joke fulfilling multiple tasks, such as setting the scene, advancing the plot, or developing a character.
+1 to Hink, the elements are the same. They still follow the three-act structure.
But often the "problem" facing the protagonists is ludicrous (a funny friendly alien needs help, the end of the world is coming, they discover a time-traveling hot tub), and their reaction to the problem is likewise ludicrous and irrational, or self-indulgent.
The essence of a joke is to lead the audience to rationally expect one outcome, and spring either the opposite or something else on them that still makes sense. "Spring" is an important word here, a joke has a surprise ending, often in the final word, and they won't laugh if they anticipated the twist. The surprise is why it is called a "punch line", it has some kind of impact like a punch. Which is why jokes wear out, people learn the punch line as one possible outcome, and stop laughing if the punch line is something they already expected. The punch line has to be unexpected to elicit a laugh.
This is actually somewhat difficult to pull off; and also difficult to analyze logically and decide whether a joke works. That takes years of experience in the comedy business. It is also why many intended comedies fall flat.
But that said, a comedy story is the same as other stories, but usually using characters with noticeably unusual personality traits, as a device by which the author(s) can create the unexpected outcomes necessary to get the audience to laugh.
For the same reason, comedy (minus stand-up) is often written in collaboration, because the author can be "too close" to their own material, they already know the punch line. So others that can be surprised by the punch line provide a valuable critique.
The key element of a humor story is whether the jokes are funny. A story is necessary, but you get much more leeway in the plot and realism of the characters and settings and reactions, most people will forgive a lot if they are busy laughing at the jokes.
In my opinion, the strongest comedy is based on perspective. The best comedians and the best humorous writers see through to the underlying comedy in situations that other people experience as serious or even tragic.
For that reason, the strongest comic storylines are ones that would work, structurally, as dramas, but that are seen through the eyes of a humorist.
Late answer but thought it needed saying;
The descriptive style of whatever happens will massively influence whether it's funny or not. This could be just my sense of humour, but IMHO the funniest books are the ones where either a) the jokes are inferred rather than described (less is usually more) and/or b) the descriptions are very unorthodox - random similes/metaphors are good for this. Take for example my favourite quote of P.G.Wodehouse;
She fitted into my biggest armchair, and made it look like it had been built around her by someone who knew they were wearing armchairs tight about the hips that season.
Granted, it would have been much more efficient and obvious to simply write "She was very fat" or "She was as fat as a ... ", but the long-drawn description used here is simply far better.
Note also that the real effort goes into describing insignificant but relatable elements - that is, things that people can relate to without becoming emotionally attached to the outcome (it becomes difficult to humourise the killing of a 'goody' in a story, to cite an extreme example, whereas uncle mike getting tangled in the hoover cord is much funnier. Unless he dies, of course...). It's that point that I disagree with @Hink's answer, because you're actually focussing on different elements, because the desired effect on the reader is different.