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
Loading your saved state.
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.

'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.
insult
'Beleidigung' and 'Frechheit' can both describe disrespectful behavior or a hurtful remark. 'Beleidigung' is more precise when someone directly puts a person down or injures their honor or dignity. 'Frechheit' is broader and often more colloquial: it can be a brazen action, an insolent reply, or disrespectful behavior, even if it is not directly an insult. Learners may confuse the words because a 'Frechheit' can also feel insulting. But not every 'Frechheit' is automatically a 'Beleidigung'.

Es war eine Frechheit, dass der Kunde einfach ohne zu bezahlen gegangen ist.
It was outrageous that the customer simply left without paying.
Die Beleidigung vor der ganzen Klasse hat ihn sehr verletzt.
The insult in front of the whole class hurt him deeply.