plague / nuisance
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plague
A great misfortune or disaster that affects many people or an entire region, often caused by a mass spread of pests or by a disease.
Die Heuschreckenplage vernichtete die gesamte Ernte des Landes.
The plague of locusts destroyed the country's entire harvest.
The noun 'die Plage' (plague/nuisance) is closely related to the verb 'plagen' (to plague/bother). Both words share the basic idea of suffering, effort, or a major annoyance.
Die Mücken sind diesen Sommer eine echte Plage.
The mosquitoes are a real nuisance this summer.
In the sense of a disaster, 'Plage' (plague) is often used for a mass increase of animals. It can also appear as an ending in compound words such as 'Heuschreckenplage' (plague of locusts), 'Mückenplage' (mosquito plague), or 'Rattenplage' (rat plague).
Eine schreckliche Dürre war die größte Plage für die Bauern.
A terrible drought was the greatest plague for the farmers.
When 'Plage' (nuisance) describes a person or thing, this is always colloquial and slightly exaggerated. The fixed expression is often 'eine (echte) Plage sein' (to be a real nuisance).
Mit seinen ständigen Fragen ist der neue Kollege eine Plage.
With his constant questions, the new colleague is a nuisance.
punishment / penalty
'die Strafe' and 'die Plage' can both colloquially mean something very unpleasant or exhausting. 'die Strafe' often sounds as if you have to endure something tedious, even though it is not necessarily a real punishment. 'die Plage' emphasizes more strongly that a person or thing is constantly annoying, disruptive, or creates a lot of work. If it is about a real consequence for wrong behavior, 'die Strafe' fits. If something is mainly a nuisance, 'die Plage' sounds more natural.
Die vielen Mücken im Garten sind im Sommer eine echte Plage.
The many mosquitoes in the garden are a real nuisance in summer.
Diese langen Formulare sind wirklich eine Strafe.
These long forms are truly real punishment.