
die Entschuldigung
apology
The expression of regret for a mistake or inconvenience that one has caused.
Ich nehme deine Entschuldigung an.
I accept your apology.
apology / excuse
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apology
The expression of regret for a mistake or inconvenience that one has caused.
Ich nehme deine Entschuldigung an.
I accept your apology.

excuse
A reason used as a justification for an omission, mistake, or absence.
Stau ist keine gute Entschuldigung, wenn man jeden Tag zu spät kommt.
Traffic is not a good excuse if you're late every day.

excuse note
A written note, especially for school or an employer, explaining an absence.
Wenn du krank bist, musst du deinem Lehrer eine Entschuldigung von deinen Eltern bringen.
If you're sick, you have to bring your teacher an excuse note from your parents.
The word consists of the prefix 'ent-' (away), the noun 'die Schuld' (guilt/blame), and the ending '-ung'. An 'Entschuldigung' (apology/excuse) is therefore meant to symbolically 'take away the guilt.'
Seine Entschuldigung klang nicht sehr aufrichtig.
His apology didn't sound very sincere.
The noun belongs to the verb 'entschuldigen' (to apologize/excuse). 'Sich für etwas entschuldigen' means to ask for forgiveness. 'Jemanden oder etwas entschuldigen' means to justify an absence or a mistake.
Er entschuldigte sich für seine Verspätung.
He apologized for being late.
When expressing an apology, people often use the phrase 'um Entschuldigung bitten' (to ask for forgiveness/to apologize). The person who receives the apology can 'annehmen' (accept) or 'ablehnen' (reject) it.
Ich möchte Sie aufrichtig um Entschuldigung für die Unannehmlichkeiten bitten.
I would like to sincerely apologize to you for the inconvenience.
Whether an apology or a justification (excuse) is meant depends on the context. The phrase 'um Entschuldigung bitten' always means an apology. 'Eine Entschuldigung haben/suchen' often suggests a justification or excuse.
Er sucht immer nach einer neuen Entschuldigung, um nicht arbeiten zu müssen.
He is always looking for a new excuse not to have to work.
In the context of school or work, 'die Entschuldigung' (excuse note) often refers to an official written note explaining an absence. In that case, you have to 'vorlegen' (present), 'abgeben' (hand in), or 'mitbringen' (bring) an excuse note.
Für das Fehlen im Unterricht braucht man eine schriftliche Entschuldigung.
For an absence from class, you need a written excuse note.
excuse
Both Entschuldigung and Ausrede can give a reason for failing to do something. An Entschuldigung can be sincere, whereas an Ausrede is often merely fabricated and does not state the true cause. Use Entschuldigung for a genuinely intended justification and Ausrede when there is such a suspicion.

Seine Entschuldigung für die Verspätung war ehrlich, weil der Bus ausgefallen war.
His excuse for being late was sincere because the bus had broken down.
Seine Erklärung für die Verspätung war nur eine Ausrede, weil er zu spät aufgestanden war.
His explanation for being late was just an excuse because he had overslept.