dragon / kite
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dragon
A large, reptile-like mythical creature that often has wings and can breathe fire.
In der Geschichte kämpft der Ritter gegen einen gefährlichen Drachen, um die Prinzessin zu retten.
In the story, the knight fights a dangerous dragon in order to save the princess.
You can recognize the meaning of 'Drache' (dragon/kite) from the context. If it is about fairy tales, fantasy, or fire, it means the mythical creature. If it is about wind, autumn, and a string, it means the toy. If someone is describing a person negatively, it means the colloquial insult.
Der Held der Geschichte muss einen Drachen besiegen.
The hero of the story has to defeat a dragon.
In its meaning as a mythical creature, 'Drache' (dragon) is often connected with the verb 'speien' (to spew/breathe), especially 'Feuer speien' (to breathe fire), or with verbs of fighting such as 'kämpfen' (to fight) or 'töten' (to kill).
Der Drache bewachte einen Schatz in seiner Höhle und konnte Feuer speien.
The dragon guarded a treasure in its cave and could breathe fire.
When referring to the toy, the fixed expression 'einen Drachen steigen lassen' (to fly a kite) is almost always used. This combination is the clearest clue to this meaning.
Am Wochenende gehen wir auf die Wiese und lassen unseren Drachen steigen.
At the weekend we're going to the meadow and flying our kite.
Using 'Drache' (dragon) for a woman is strongly colloquial and derogatory. It is an insult and should, if used at all, be used only with great caution.
Er nannte seine Schwiegermutter hinter ihrem Rücken einen Drachen.
He called his mother-in-law a dragon behind her back.
monster
'das Monster' and 'der Drache' can both be frightening creatures from stories, fairy tales, or films. 'Monster' is more general: it can have many different forms and is often ugly, large, or dangerous. 'Drache' is more specific: people usually imagine a reptile-like creature with wings that can often breathe fire. If the type of creature is not important, 'Monster' fits. If you mean this typical mythical creature, 'Drache' is more precise.
Der Drache bewacht im Märchen einen Schatz in der Höhle.
In the fairy tale, the dragon guards a treasure in the cave.
Im Film kommt nachts ein Monster aus dem Wald.
In the film, a monster comes out of the forest at night.