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.

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.
justification / founding
'die Begründung' and 'die Entschuldigung' can both explain why something happened or why someone did not do something. A 'Begründung' is more neutral and fits decisions, rules, or actions. An 'Entschuldigung' often has to do with a mistake, being late, or an absence. It can also mean that someone expresses regret, or be a written note for school or work. It is easy to confuse the words when giving a reason. If you are simply explaining factually why something is the way it is, 'Begründung' fits. If it is about a mistake, an absence, or saying 'I'm sorry', 'Entschuldigung' fits.

Ich bringe morgen eine Entschuldigung mit, weil ich heute beim Unterricht gefehlt habe.
I'll bring an excuse note tomorrow because I missed class today.
Für die neue Regel gibt die Schule eine kurze Begründung.
The school gives a short justification for the new rule.