
die Klage
complaint
The expression of dissatisfaction, pain, or grief about something. A complaint or lament.
Seine ständige Klage über die Arbeit nervt die Kollegen.
His constant complaining about work annoys his colleagues.
complaint / lawsuit
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complaint
The expression of dissatisfaction, pain, or grief about something. A complaint or lament.
Seine ständige Klage über die Arbeit nervt die Kollegen.
His constant complaining about work annoys his colleagues.

lawsuit
A formal legal action in which a person or organization sues another in court in order to enforce a claim. A synonym is 'Anklage' (charge/indictment) in criminal law or 'Klageerhebung' (filing a lawsuit) in civil law.
Das Unternehmen hat gegen den Konkurrenten eine Klage wegen Patentverletzung eingereicht.
The company filed a lawsuit against its competitor for patent infringement.
'Die Klage' (complaint/lawsuit) is the noun related to the verb 'klagen' (to complain/to sue). The preposition used with the verb helps you understand the meaning: 'über etwas klagen' (to complain about something) leads to 'Klage' as a complaint, while 'gegen jemanden klagen' (to sue someone) leads to 'Klage' as a court case.
Er klagt ständig über das Wetter.
He constantly complains about the weather.
You can recognize the difference between the meanings from the context. Words like 'Gericht' (court), 'Anwalt' (lawyer), or 'einreichen' (to file) point to the legal meaning. Words like 'Schmerz' (pain), 'Leid' (suffering), or 'ständig' (constant) point to the complaint meaning.
Ihre Klage über Kopfschmerzen wurde immer lauter.
Her complaint about headaches grew louder and louder.
When 'Klage' means a complaint, the reason for it is expressed with the preposition 'über' and the accusative.
Die Klage über den Lärm der Baustelle war berechtigt.
The complaint about the noise from the construction site was justified.
In a legal context, 'Klage' is often used with the verbs 'einreichen' (to file) or 'erheben' (to bring/file). The person or institution being sued is named with the preposition 'gegen' (against).
Der Verein hat Klage gegen die Entscheidung der Stadt eingereicht.
The association filed a lawsuit against the city's decision.
complaint / ailment
Klage and Beschwerde can both be an expression of dissatisfaction. Klage tends to emphasize expressing pain or suffering in a plaintive way, while Beschwerde usually means specific criticism of a thing, service, or person. Use Beschwerde for a targeted complaint and Klage for expressing distress or suffering.

Seine Klage über die lange Wartezeit klang vor allem traurig und erschöpft.
His complaint about the long wait sounded above all sad and exhausted.
Er schrieb eine Beschwerde über die lange Wartezeit an den Kundenservice.
He wrote a complaint about the long wait to customer service.