die Verletzung
injury
Physical damage or a wound caused by an accident, violence, or an illness.
Nach dem Sturz hatte er eine schwere Verletzung am Knie.
After the fall, he had a serious injury to his knee.

injury / violation
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injury
Physical damage or a wound caused by an accident, violence, or an illness.
Nach dem Sturz hatte er eine schwere Verletzung am Knie.
After the fall, he had a serious injury to his knee.

'Die Verletzung' (injury / violation) is the noun form of the verb 'verletzen' (to injure / to violate / to hurt). The verb can likewise mean physical 'Verwunden' (wounding), breaking rules ('missachten' — to disregard), or hurting feelings.
Pass auf, dass du dich an dem kaputten Glas nicht verletzt.
Be careful that you don't injure yourself on the broken glass.
The context makes clear which meaning is intended. If it is about the body (knee, head), it means a wound. If it is about rules (law, contract), it means a violation. If it is about feelings (pride, honor), it means an offense or hurt.
Die Verletzung am Bein heilte nur langsam.
The injury to his leg healed only slowly.
In connection with physical wounds, you often hear the verbs 'sich eine Verletzung zuziehen' (to sustain an injury) or 'eine Verletzung erleiden' (to suffer an injury). Typical adjectives are 'schwer' (serious), 'leicht' (minor), or 'lebensgefährlich' (life-threatening).
Der Fußballspieler hat sich im Training eine leichte Verletzung zugezogen.
The soccer player sustained a minor injury during training.
When 'Verletzung' (violation) means a breach of rules, it is often followed by a noun in the genitive that names the rule that was broken. Common examples are 'Verletzung des Gesetzes' (violation of the law), 'Verletzung der Menschenrechte' (violation of human rights), or 'Verletzung der Privatsphäre' (violation of privacy).
Das unbefugte Betreten des Grundstücks ist eine Verletzung des Hausrechts.
Unauthorized entry onto the property is a violation of the householder's rights.
damage / harm
'der Schaden' and 'die Verletzung' can both mean that something negative happens to a person. 'Verletzung' is more specific when the body is affected, for example a wound or a broken arm. Feelings can also be hurt. 'Schaden' is broader: it can affect health, money, reputation, or an object. You can say that someone 'zu Schaden kommt' (comes to harm) when they are injured or negatively affected. But when you are talking about a specific wound, 'Verletzung' is more natural.

Nach dem Sturz hat sie eine Verletzung am Knie.
After the fall, she has an injury to her knee.
Zum Glück ist bei dem kleinen Brand niemand zu Schaden gekommen.
Fortunately, no one came to harm in the small fire.