When it comes to expressing embarrassment in German, it's important to not only know the relevant adjectives and verbs, but understand exactly how they are used. 

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For example, there are the adjectives peinlich and verlegen. These both describe uncomfortable emotional states in German, but they have different meanings and uses. The adjective peinlich refers to a situation or behavior that causes shame or embarrassment, often in a social context.

Das ist so peinlich! Erst lade ich dich ein und dann passiert so was.

This is so embarrassing! First, I offer to pay for you and then something like this happens.

Caption 22, Nicos Weg: Partnersuche

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Das muss dir doch nicht peinlich sein, wenn dir meine Musik gefällt.

You don't have to be embarrassed if you like my music.

Caption 23, Die Pfefferkörner: Das Wunderkind

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Es war ihr voll peinlich, als ich sie erwischt habe.

She was totally embarrassed when I caught her.

Caption 39, Die Pfefferkörner: Gerüchteküche

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The adjective verlegen describes the internal state of a person who feels uncomfortable, shy, or uncertain, corresponding to the English word "embarrassed." 

Junge Frau ... Sie machen mich ganz verlegen.

Young lady... You're making me feel quite embarrassed.

Caption 25, Nicos Weg B1: Richtiges Benehmen

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Von Verlegenheit überhaupt keine Spur, er ist ene wahre Frohnatur

No trace of embarrassment at all, he is truly a good-natured guy

Caption 14, Herbert Grönemeyer: Was soll das?

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So, while a situation can be peinlich, a person is typically verlegen as a reaction to it. It's important to note that verlegen is also a verb, and it has a completely unrelated meaning! You can read about the distinction here. 

When it comes to embarrassing oneself rather than feeling embarrassed, the correct verb is sich blamieren. 

...denn dieses Verhalten kann in der Regel nur zum Scheitern führen oder dazu, dass man sich blamiert.

... because this behavior can, as a rule, lead to failure or to you embarrassing yourself.

Captions 55-56, Eva erklärt: Sprichwörter

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When it comes to embarrassing someone else, another structure to know is in Verlegenheit bringen, as in Er wollte sie damit nicht in Verlegenheit bringen.

Further Learning
You'll find more examples of these adjectives and verbs on Yabla German. 

The referendum in Great Britain to leave the European Union was nicknamed "Brexit," a portmanteau of the words "Britain" and "exit." The same nickname was used in Germany. Let's take a moment to divert our attention from that event and focus instead on the different ways we can say "exit" in German. 

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Wir können die nächste Ausfahrt nehmen oder noch weiterfahr'n

We can take the next exit or keep on driving

Caption 7, Deutsche Bands - Luxuslärm

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Die Ausfahrt is the most commonly seen translation of "exit" and is usually used in reference to the off-ramp of a roadway.

Wo ich reinkomm', geht man raus.

Where I enter, they exit.

Caption 43, Frank Zander - Hier kommt Kurt

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This is "exit" in its simple slang form of rausgehen, or "to go out."

Und zeigst nur stumm auf die Ausgangstür

And just point silently to the exit door

Caption 10, Herbert Grönemeyer - Was soll das?

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Der Ausgang is usually meant as the point where you leave a building.

Brüller [Umgangssprache], Lacher, Brüller, Abgang,

Big laugh, laugher, big laugh, exit,

mehr ist net [Dialekt, nicht] zu machen.

there isn't more to do.

Caption 19, Ab durch die Heimat - 4 Comedians unterwegs im Südwesten

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Der Abgang is the simple act of leaving or the act of leaving a specific sphere of activity. It can also mean the person who has left, as in "the one who left," although the more common terms are der Abgänger or die Abgängerin. Der Abgang is still occasionally used in medical and military circles to refer to death.

Das sind alles Leute, die eigentlich nichts weiter verbrochen haben,

These are all people who actually violated nothing more

als dass sie vielleicht einen Ausreiseantrag gestellt.

than that they perhaps submitted an exit visa application.

Captions 27-28, DDR zum Anfassen - Ganz tief im Westen

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Der Ausreiseantrag or "exit visa application" uses the word die Ausreise for "exit," which on its own would usually be translated as "departure." Hopefully the citizens of the United Kingdom —possibly soon consisting only of non-EU countries England and Wales — will not be required to file an Ausreiseantrag in order to leave what remains of the UK. It is nearly certain, however, that they will now have to get a residence permit (die Aufenthaltserlaubnis) to live in the EU. 

Further Learning
Do a search on Yabla German to see the different ways that terms for "exit" such as der Abgang, die Ausfahrt, and der Ausgang are used in a real world context.