schädigen
to damage
To cause harm to someone or something; to injure, damage, or impair the function of something or someone.
Rauchen schädigt die Lunge und das Herz-Kreislauf-System.
Smoking damages the lungs and the cardiovascular system.

to damage / harm
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to damage
To cause harm to someone or something; to injure, damage, or impair the function of something or someone.
Rauchen schädigt die Lunge und das Herz-Kreislauf-System.
Smoking damages the lungs and the cardiovascular system.

The verb 'schädigen' derives directly from the noun 'der Schaden' (damage/harm). It literally means to cause harm or damage to someone or something.
Autoabgase schädigen die Umwelt.
Car exhaust fumes damage the environment.
'Schädigen' is a transitive verb and always requires an accusative object. So the structure is: someone/something (subject) damages someone/something (accusative object).
Zu laute Musik kann das Gehör schädigen.
Music that is too loud can damage your hearing.
In the literal meaning, 'schädigen' is often combined with body parts or the environment. Typical collocations are 'die Gesundheit schädigen' (to damage one's health), 'die Nieren schädigen' (to damage the kidneys), or 'das Ökosystem schädigen' (to damage the ecosystem).
Der Konsum von zu viel Zucker schädigt die Zähne.
Consuming too much sugar damages the teeth.
In the figurative meaning, 'schädigen' refers to abstract things. Common combinations are 'den Ruf schädigen' (to damage someone's reputation), 'das Ansehen schädigen' (to damage someone's standing), or 'das Image einer Firma schädigen' (to damage a company's image).
Falsche Gerüchte können den Ruf einer Person ernsthaft schädigen.
False rumors can seriously damage a person's reputation.
to injure / hurt / violate
'verletzen' and 'schädigen' both mean that something negative happens to someone or something. 'verletzen' is especially used for physical wounds and also for feelings. You can injure your knee or hurt someone with words. 'schädigen' sounds more general and often more factual. It fits when damage or harm is caused, for example to health, the environment, a company, or someone's reputation. Learners may confuse the words because both refer to harm. As a simple guide: for wounds and hurt feelings, 'verletzen' is more natural; for general damage, harm, or disadvantages, 'schädigen' is more precise.

Zu viel Lärm kann auf Dauer die Gesundheit schädigen.
Too much noise can damage your health in the long run.
Beim Fußball hat sich Max am Fuß verletzt.
Max injured his foot while playing soccer.