
die Frechheit
impudence
A quality or behavior that is perceived as insolent, disrespectful, or brazen.
Seine Frechheit, einfach vorzudrängeln, hat alle geärgert.
His impudence in simply pushing to the front annoyed everyone.
impudence / cheek
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impudence
A quality or behavior that is perceived as insolent, disrespectful, or brazen.
Seine Frechheit, einfach vorzudrängeln, hat alle geärgert.
His impudence in simply pushing to the front annoyed everyone.

outrage
A single action or statement that is considered insolent or brazen.
Was du da sagst, ist eine absolute Frechheit!
What you're saying there is an absolute outrage!
'Die Frechheit' (impudence/cheek) is the noun formed from the adjective 'frech' (cheeky/impudent). The ending '-heit' is often used to form a noun from an adjective that describes the corresponding quality.
Sein freches Grinsen war schon eine kleine Frechheit.
His cheeky grin was already a bit of an impudence.
The word can refer to a general quality (uncountable) or to a single insolent action (countable). When a single act is meant, it is often used with the indefinite article 'eine' or in the plural.
Seine Frechheit kannte keine Grenzen.
His impudence knew no bounds.
A very common and strong phrase is 'die Frechheit besitzen, etwas zu tun' (to have the nerve to do something). It expresses great indignation at a person's brazenness.
Er besaß die Frechheit, mich nach allem um einen Gefallen zu bitten.
He had the nerve to ask me for a favor after everything.
impudence / outrage
Both words can refer to a disrespectful remark or action. Frechheit is usually more colloquial and can sound somewhat less severe, whereas Unverschämtheit emphasizes the perceived imposition and violation of respect more strongly. Use Frechheit for the milder judgment and Unverschämtheit for particularly brazen behavior.

Dass du meinen Stift einfach nimmst, ist eine Frechheit.
It's outrageous that you just take my pen.
Dass du mich vor allen beleidigst, ist eine Unverschämtheit.
It's outrageous that you insult me in front of everyone.