
der Einfall
idea
A sudden thought or spontaneous idea that comes to someone's mind.
Mir kam gerade ein genialer Einfall, wie wir das Problem lösen können.
I just had a brilliant idea for how we can solve the problem.
idea / invasion / incidence
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idea
A sudden thought or spontaneous idea that comes to someone's mind.
Mir kam gerade ein genialer Einfall, wie wir das Problem lösen können.
I just had a brilliant idea for how we can solve the problem.

invasion
The hostile entry of troops into foreign territory; an invasion.
Der Einfall der gegnerischen Armee in das Land erfolgte im Morgengrauen.
The invasion of the country by the opposing army took place at dawn.

incidence
In physics, the striking of rays, such as light or sound, on a surface.
Beim Spiegel ist der Winkel des Einfalls gleich dem Winkel der Reflexion.
With a mirror, the angle of incidence is equal to the angle of reflection.

collapse
The caving in or collapse of something, for example in a mine or in a building.
Der Einfall des alten Stollens blockierte den Hauptweg für die Bergleute.
The collapse of the old tunnel blocked the main path for the miners.
'Der Einfall' (idea/invasion/incidence/collapse) is the noun form of the separable verb 'einfallen'. The different meanings of the noun correspond directly to the meanings of the verb: an idea occurs to someone, troops invade a country, or a building collapses.
Plötzlich fiel ihr die Lösung für das Rätsel ein.
Suddenly the solution to the puzzle occurred to her.
You can recognize the meaning from the context: With creativity and problems, it is about an idea. With military and history, it is about an invasion. With physics and optics, it is about rays striking a surface. With mining and buildings, it is about a collapse.
Der plötzliche Einfall von Kaltluft führte zu einem Wetterumschwung.
The sudden influx of cold air led to a change in the weather.
When a person has an idea, the phrase 'einen Einfall haben' (to have an idea) is often used. The related verb 'einfallen' needs the dative: 'Mir fällt etwas ein.' (Something occurs to me.) Not: 'Ich falle etwas ein.'
Ich hatte gestern den besten Einfall für dein Geburtstagsgeschenk.
Yesterday I had the best idea for your birthday present.
The most common meaning, 'Idee' (idea), is often combined with adjectives like 'gut' (good), 'genial' (brilliant), 'plötzlich' (sudden), or 'lustig' (funny). A fixed phrase is 'einen Einfall haben' (to have an idea).
Das war wirklich ein lustiger Einfall von dir, eine Kostümparty zu machen.
That was really a funny idea of yours, having a costume party.
The meaning 'Invasion' (invasion) is limited to military or historical contexts. It describes the hostile entry of armed groups into a territory.
Das Museum dokumentiert den Einfall der Truppen im Jahr 1944.
The museum documents the invasion of the troops in 1944.
This meaning is a technical term from physics, specifically optics. It is used almost exclusively in scientific or school texts to describe rays striking a surface.
Im Physikunterricht haben wir den Einfall von Licht auf eine Linse untersucht.
In physics class, we investigated the incidence of light on a lens.
idea
Idee refers to a thought for a solution or a design without saying how it came about. Einfall emphasizes that the thought occurs suddenly or spontaneously. If this surprising emergence is important, Einfall fits; otherwise, Idee is the more neutral choice.

Beim Vorbeigehen hatte Mira plötzlich den Einfall, in die leere Ecke ein Regal zu stellen.
As she walked past, Mira suddenly had the idea of putting a shelf in the empty corner.
Nach längerer Planung hatte Mira eine Idee für die leere Ecke.
After some planning, Mira had an idea for the empty corner.