
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.

to damage
To harm the reputation, standing, or interests of someone or something.
Der Skandal hat den Ruf des Unternehmens nachhaltig geschädigt.
The scandal permanently damaged the company's reputation.
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.
damage / harm
Schaden is the disadvantage that has resulted, for example to a person’s reputation. Schädigen, as a verb, describes causing this disadvantage. When naming the result, use Schaden; when expressing the negative effect of an action, use schädigen.

Die falsche Meldung schädigte seinen Ruf erheblich.
The false report seriously damaged his reputation.
Die falsche Meldung fügte seinem Ruf großen Schaden zu.
The false report caused serious damage to his reputation.