Anvilicious: The film concludes with a 4 1/2 minute speech delivered directly to the camera. While Chaplin's political views were complicated, the clear takeaway is "Down with Hitler."
Big-Lipped Alligator Moment: After the Barber and Hannah escape to the roof while running away from the soldiers, there is a short scene of Hynkel playing the piano, after which it goes back to the Barber and Hannah on the roof. The piano shot serves no purpose (except maybe for Charlie to show off) and is just kind of there.
"Funny Aneurysm" Moment: Nearly every scene after the WWI sequence, especially the concentration camp sequence and the scenes of Hynkel killing his men. Chaplin himself said that if he had known the extent of the Nazis' "homicidal insanity," then he could not have made the film.
Genius Bonus: The signs in the Jewish Quarter are written in Esperanto. In Mein Kampf, Hitler wrote that he thought it was created as a tool to unify the Jewish diaspora.
Harsher in Hindsight: Chaplin only had a faint picture of the atrocities occurring in Nazi Germany when the film was being produced, based on what refugees from Europe had told him. Back in 1940, the worst things have not started yet anyway. He would later state that had he known the full extent of the Nazis' crimes, he would never have made the film.
The concentration camp scenes in particular make for quite awkward viewing. Perhaps the most In Your Face example of this trope is when the Jewish Barber first arrives at the camp. Whereas all the prisoners walk in one direction, he walks toward another. When one of the guards asks him where he's going, he cheerfully answers: "The Smoking Room," before being pushed the other way. In Auschwitz, prisoners were pointed to walk in one direction or another, and people waved in the wrong direction got sent to gas chambers.
Hynkel makes a stirring speech about Tomania's expansion, including France, Finland and Russia. France had already been invaded by Nazi Germany, Russia would be in the next year, and Finland would remain Germany's co-belligerent during the latter.
Heartwarming in Hindsight: With the allegations of Nazi Germany's influence on Hollywood during its golden age coming to light, Chaplin's determination to produce an anti-Nazi satire feature film in defiance of it comes off as downright heroic.
Memetic Mutation: For some reason the Rousing Speech has been seeing some spread, in the form of clips like this one. Such clips are rather frequently posted under the title "The Greatest Speech Ever Made", or some variant of it.
Some Anvils Need to Be Dropped: The anti-Nazi message isn't exactly subtle but this was the US of 1940 and a large minority in the US had some sympathy for or were impressed by the Nazi regime. Effective, easy-to-understand and revealing satire of Nazism and fascism was badly needed.
Stock Phrase: "Schtonk!" alone stands for the whole film (up to the point that the German film parody of the fake Hitler diary affair is called this way).
Values Dissonance: Chaplin's speech really doesn't fit with a modern understanding of fascism. He describes the Nazis as "machine men with machine minds" who "think too much and feel too little." It's now widely accepted that fascism is a highly emotional and irrational movement, generally brought about by paranoia and xenophobia.
Or it can be just as easily interpreted as talking about "machines" behind dictatorship in general: war and propaganda, as they turn humans into nothing more than nameless cogs.