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.