So comedy in Nazi Germany existed, but surprisingly enough, it was usually directed at the Nazis themselves. Germans were aware of the political climate, and while most supported Hitler the man, many regular Germans hated the actual Nazi party since they were viewed as corrupt or self serving. Some post war historians and Germans have attempted to say that Germans "passively resisted" the Nazis through jokes, but the truth is that these were more just to blow off steam and were not intended to be taken as some sort of serious political statement. Generally as long as one kept their jokes private, as in among friends, the Nazis wouldn't really make a huge deal about it. The issue came when you got denounced or brought before the infamous "people's courts". The penalty for telling a few jokes would be a minor slap on the wrist, as long as the joke wasn't directly mocking Hitler or the Nazi state. Of course this didn't apply if you were considered a threat to Nazi control; like if you got caught telling a joke about the Nazis and then it was discovered you were a member of the Communist party, you would be up shit creek without a paddle. As the war dragged on the Nazis became ever more brutal in defending their cult of personality, and the penalties for any sort of dissent, including joke telling, got far more severe.

A Nazi police officer writing in 1937 outlined the Nazi position on jokes:

For some time the devising and telling of political jokes has grown to become a real nuisance. So long as these jokes are the expression of a sound spirit and are harmless in character, there will be, as has been repeatedly underlined at the top level of government, nothing to object to in them. But if they are slanderous in content, then for security reasons we can and must not tolerate their being spread around.

These "whispered jokes" were aimed more at the Nazis themselves, and not their crimes or the actual state. For example here is a popular joke about Herman Goering that pokes fun at his weight and his outlandish outfits:

Göring recently added an arrow to the many medals on his chest. It’s there as a directional sign: “To be continued on my back."

Another popular joke aimed at the Winter Relief Fund, which was a Nazi charity that helped the poor during the winter:

A poster for the Winter Relief Fund reads: “No one should be allowed to go hungry or suffer from the cold.” A workersays to a colleague: “So now we’re not even allowed to do that.”

German Jews actually developed their own, darker form of comedy. While ordinary Germans told jokes as a way of blowing off steam, you can still sense they love their country and that they generally support the Nazis. German Jewish humour has a distinctive tone to it. It was intended to breed hope in the face of the massive amounts of discrimination that the Jews faced. A popular joke among the few Jews left in Germany near the end of WW2 goes as follows:

Two Jews are waiting to face a firing squad, when the news arrives that they are to be hanged instead. One turns to the other and says: “You see—they’ve run out of ammunition!”

The joke here is that the Jews are finally catching a break, and that there is still hope for the Jews because clearly the Nazis are not doing to well if they are running out of ammunition. Another joke lampoons the Nazi uniforms:

A Swiss visiting a Jewish friend in the Third Reich asks him: “So how do you feel under the Nazis?” He answers: “Like a tapeworm. Every day, I wriggle my way through a mass of brown stuff and wait to be excreted.”

Levi and Hirsch bump into one another in the wilderness of Sudan. Each of them is carrying a heavy rifle and leading a column of bearers. “How it going,” asks one. “What are you doing here?” “I’ve got an ivory-carving shop in Alexandria, and to keep costs down, I shoot the elephants myself. And you?” “Much the same. I've got a crocodile leather business in Port Said and am here hunting for crocs.” “And what’s the story with our friend Simon?” “Oh, he’s a real adventurer. He stayed in Berlin.”

A number of prewar German Jewish comedians died in Hitler's camps. A Jewish German compiler of these jokes by the name of Salcia Landmann explained the nature and the reasoning behind these seemingly dark jokes:

as an expression of Jews’ will to survive against all odds. These jokes make fun of the terrors Jews experienced every day. As such, the blackest Jewish humour expresses defiance: I laugh, therefore I am. My back is to the wall, and I’m still laughing.

Now the Nazis weren't really big on comedy, but they did have a few moments where they tried to "lighten the mood" at some of their rallies. For example the Hitler Youth once put on a skit that lampooned pre Nazi politics in Germany:

“What moved people in Germany, what made them split their sides laughing, was the musty old Wiemar Republic and its democracy. The entire public shared this view, and they rolled on the door giggling. On the evening in question, the skit was set in parliament. One of the boys, dressed as a parliamentary deputy, slept the entire time. When suddenly an alarm clock went off, the audience cried with delight. Another deputy had a speech impediment; a third, a problem with flatulence. The entire evening was devoted to making fun of a democracy that had passed its sell-by date, and the audience couldn't get enough.

Now another popular category for jokes among non Jewish Germans, was political humour, this was a far more dangerous form of comedy since it involved mocking not only the Nazis state, but mocking the character and the actions of top Nazis, including Adolf Hitler. For example here is a joke directed at the Nazi policy of labelling those who opposed them as "radicals" or "reactionaries":

Question: What is a reactionary? Answer: Someone who occupies a wellpaying job coveted by a Nazi.

A cook is trying to make fried potatoes without lard and begins waving a swastika banner over the oven. When asked why, she answers: “Under this flag, a lot of lard-asses seem to have come out pretty well.”

This joke pokes fun at the fact that in German, the word to dial someone on the phone also means vote in an election:

The telephone rings, and a man says: “Hello, can I speak to Müller?” “Who?” “Müller. Is Müller there?” “No, my name is Schmidt.” “Oh, I’m sorry, I must have dialed the wrong person.” “No big deal, we all did that in the last election.”

These two jokes poke fun at the Nazis insentience that people use the greeting "Heil Hitler":

Hitler visits a lunatic asylum, where the patients all dutifully perform the German salute. Suddenly, Hitler sees one man whose arm is not raised. “Why don’t you greet me the same way as everyone else,” he hisses. The man answers: “My Führer, I’m an orderly, not a madman!”

Tünnes and Schäl are walking across a cow pasture, when Tünnes steps in a mound of cowshit and almost falls down. Immediately he raises his right arm and yells, “Heil Hitler!” “Are you crazy?” asks Schäl. “What are you doing? There’s no one else around here.” “I’m following regulations,” Tünnes answers. “Whenever you step into anywhere, you’re supposed to say ‘Heil Hitler.’ ”

This joke makes reference to the Nazi policy of subjugating the judicial system to their own needs:

In Switzerland a Nazi bigwig asks the purpose of a public building. ‘That’s our Ministry of Marine,’ says the Swiss man. The Nazi laughs and mocks him. ‘You with your two or three ships, what do you need a Ministry of Marine for?’ The Swiss man: ‘Yes, - so what do you still need a Ministry of Justice in Germany for then?’

Ultimately humour in Nazi Germany functioned as an escape, whether you were a Jew in a concentration camp or a regular German citizen, humour was there to help you make light of the an increasingly bleak society. It helped inject a bit of light into what seemed like dark, serious situations.

The best source on humour both in and about Nazi Germany is a book called "Dead Funny: Telling Jokes in Hitler's Germany' by Rudolph Herzog and the book "The Third Reich in Power" by Richard J. Evans also touches on the subject slightly, but it is a very small part of the book.

Since you mentioned film and /u/Sid_Burn hasn't yet covered those in his excellent reply, the Nazis didn't only use them for propaganda. Especially during the later phase of the War, the Nazis recognized the medium's value in providing an escape from the often harsh reality, so they produced movies that were entirely apolitical and designed to keep up morale through distraction, including comedy.

The most prominent example of this is [Die Feuerzangenbowle](http://en.wikipedia.org/wiki/Die_Feuerzangenbowle_(1944_film)) (The Punch Bowl), which might still be the funniest German comedy film to date and worth watching if you can find subtitles (OT version is available via Archive.org, link on Wikipedia page).

There were things like Die Feuerzangenbowle, a comedy film released in 1944, which ridicules the institution of school, teachers, and by extension authority and conservative societal order. While it had to be approved by the authorities, it has no propaganda content - it actually makes no reference to the Nazis or the war at all and could be set in the Weimar Republic or during the German Empire. The filmmakers even drew out the shoot to keep actors from being drafted, and apparently things like that were possible despite the state's tight grip.

It's also hilarious, great film-making considering the time, and I highly recommend to watch it.

Edit: added a comma.

Claire Waldorf was one of Germany's most popular cabaret singers and comedians during the Weimar years and into the early Nazi era. She got in trouble with a song called "Hermann heeßt er" which pokes fun of Hermann Göring's weight.

„Rechts Lametta, links Lametta, und der Bauch wird immer fetta und in Preußen ist er Meester – Herrmann heeßt er!

Translated into English that means something like this: „Tinsel to the right, tinsel to the left, and the stomach’s growing fat, and in Prussia he is master, Hermann he was called there.” Hermann Göring forbid Claire Waldorff to sing the song even though she had not even texted the additional verse. In Berlin it became hard for her to get an engagement after 1936, but until 1943 her appearances can be proved, in January 1942 even in the occupied Paris. After the war Claire Waldorff lived together with her girlfriend Olga von Roeder quite secluded, she died in 1957 aged 72.

Source: http://20onemagazine.com/2013/07/18/forgotten-victims/

Hear the song: http://youtu.be/1j1rqxtyT3o