
die Geschichte
story
A narrated or written sequence of events that may be true or invented; a narrative.
Kannst du mir eine Geschichte zum Einschlafen vorlesen?
Can you read me a bedtime story?
story / history
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story
A narrated or written sequence of events that may be true or invented; a narrative.
Kannst du mir eine Geschichte zum Einschlafen vorlesen?
Can you read me a bedtime story?

history
The totality of past events, especially those of humanity; also the academic discipline that deals with this.
Die Berliner Mauer ist ein wichtiges Symbol der deutschen Geschichte.
The Berlin Wall is an important symbol of German history.

matter / affair
Colloquially, a matter, event, or incident, often with a complicated or negative connotation.
Die ganze Geschichte mit dem verlorenen Schlüssel war sehr unangenehm.
The whole affair with the lost key was very unpleasant.
'Geschichte' (story/history) is closely related to the verb 'geschehen' (to happen, to occur). So a 'Geschichte' is literally something that has 'geschehen' (happened).
Was ist denn hier geschehen?
What happened here?
The context makes it clear whether 'story' or 'history' is meant. If it is about personal narratives, books, or films, it means 'story'. If it is about the past of countries or the academic field, it means 'history'.
Er hat mir die ganze Geschichte von seinem Urlaub erzählt.
He told me the whole story about his vacation.
When 'Geschichte' (story) means a narrative, it is often combined with verbs such as 'erzählen' (to tell), 'schreiben' (to write), 'erfinden' (to invent), or 'vorlesen' (to read aloud).
Mein Opa kann sehr gut Geschichten erzählen.
My grandpa is very good at telling stories.
In everyday language, 'die Geschichte' (the matter/affair) can also refer to a complicated or unpleasant matter. This use is informal.
Die Geschichte mit dem kaputten Auto hat viel Geld gekostet.
The affair with the broken car cost a lot of money.
past
Geschichte focuses on past events and how they developed. Vergangenheit refers more to the period or stage of life that has passed. Someone who tells what happened earlier speaks of Geschichte; someone who means an earlier period as a distinct stage speaks of Vergangenheit.

Das Museum zeigt die wechselvolle Geschichte des Landes.
The museum presents the country's turbulent history.
Viele Konflikte des Landes liegen weit in der Vergangenheit.
Many of the country's conflicts lie far in the past.