Definition
The main German word for “brain” is Gehirn or just Hirn. This word is thought to be related to “horn” (in German, Horn), in the sense of ‘“uppermost part of the body”, and as such is also related to “hornet” (Hornisse), because of the stinger, and “hart” (a male deer, Hirsch), because of the antlers.
In the sense of “reason” or “intellect”, the word for “brain” in German is Verstand. This word also means “mind”, although Geist (which also means “spirit” or “ghost”) is slightly more common.
Kopf, meaning “head” can also be used to mean “brain”.
Examples
Benutz dein Gehirn! Use your brain!
Er hat seinen Verstand verloren. He’s lost his mind.
Der Geist ist willig, aber das Fleisch ist schwach. The spirit is willing but the flesh is weak.
Sie hatte eine Gehirnerschütterung. She’s had a (brain-)concussion.
Ich habe mir die ganze Nacht den Kopf zerbrochen. I racked my brain all night.