der Geist
mind
A person's ability to think, feel, and will; the reason or intellect.
Sie ist eine Frau von wachem Geist und großem Herzen.
She is a woman with an alert mind and a big heart.

spirit / mind
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mind
A person's ability to think, feel, and will; the reason or intellect.
Sie ist eine Frau von wachem Geist und großem Herzen.
She is a woman with an alert mind and a big heart.

The plural 'die Geister' almost always refers to ghosts or supernatural beings. The other meanings of 'Geist' (mind, essence, alcohol) are usually used in the singular.
Glaubst du an Geister?
Do you believe in ghosts?
The word family of 'Geist' (spirit/mind) helps with understanding. Adjectives like 'geistig' (mental) or verbs like 'begeistern' (be-geistern, literally: to infuse with a spirit) relate to the mind or soul.
Regelmäßiges Lesen fördert die geistige Fitness.
Regular reading promotes mental fitness.
When 'Geist' (mind) refers to the intellect, it is often described with adjectives like 'scharf' (sharp), 'wach' (alert), 'klar' (clear), or 'brillant' (brilliant).
Trotz ihres hohen Alters hat sie noch einen sehr wachen Geist.
Despite her advanced age, she still has a very alert mind.
To describe the 'Essenz' (essence) or 'Charakter' (character) of something, the structures 'im Geist von...' (in the spirit of...) or 'der Geist der/des...' (the spirit of the...) are often used. A well-known example is the word 'Zeitgeist'.
Die Entscheidung wurde ganz im Geist der Unternehmensphilosophie getroffen.
The decision was made entirely in the spirit of the company's philosophy.
The expression 'seinen Geist aufgeben' (to give up the ghost) is used colloquially for devices that break down. The possessive pronoun ('sein', 'ihr') agrees with the device.
Meine alte Waschmaschine hat gestern endgültig ihren Geist aufgegeben.
My old washing machine finally gave up the ghost yesterday.
In the context of drinks or distilleries, 'Geist' (spirit) refers to an alcoholic spirit obtained through distillation. It often appears in compound words such as 'Himbeergeist' (raspberry spirit).
Als Digestif bestellten wir einen Obstler, einen klaren Geist aus Äpfeln und Birnen.
As a digestif, we ordered an Obstler, a clear spirit made from apples and pears.
head
'der Kopf' and 'der Geist' can both have to do with thinking. 'Kopf' sounds very natural in everyday language when talking about thoughts, concentration, or reason. 'Geist' is more abstract and means not only thinking, but also feeling and will. It often sounds somewhat more elevated or general. The words can be confused because both describe not only the body, but also a person's inner life. In normal everyday sentences, people often say 'im Kopf' (in one's head); when talking about mental ability or inner strength, 'Geist' is usually a better fit.

Regelmäßiges Lesen hält den Geist wach.
Regular reading keeps the mind alert.
Nach dem Gespräch habe ich die ganze Zeit diese Frage im Kopf.
After the conversation, I had that question in my head the whole time.