inner
Refers to a person's thoughts, feelings, or psyche.
Trotz des Lärms fand er zu seiner inneren Ruhe zurück.
Despite the noise, he found his way back to his inner calm.

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Refers to a person's thoughts, feelings, or psyche.
Trotz des Lärms fand er zu seiner inneren Ruhe zurück.
Despite the noise, he found his way back to his inner calm.

The adjective 'inner' is used almost exclusively before a noun (attributive use). It usually does not stand alone at the end of a sentence.
Er spürte einen tiefen inneren Frieden.
He felt a deep inner peace.
'Inner' is derived from the preposition 'in' (in). The ending '-er' is often used to form adjectives from expressions of place, for example also with 'außen' (outside) → 'äußerer' (outer) or 'oben' (above) → 'oberer' (upper).
Der innere Teil der Mauer muss repariert werden.
The inner part of the wall needs to be repaired.
In the psychological sense, 'inner' is often combined with feelings and states. Common phrases are 'innere Ruhe' (inner calm), 'innerer Konflikt' (inner conflict), 'innere Stimme' (inner voice), or 'innere Uhr' (internal clock).
Manchmal muss man einfach auf seine innere Stimme hören.
Sometimes you simply have to listen to your inner voice.
When referring to a location, 'inner' is often found in fixed terms. Typical examples are 'innere Organe' (internal organs) in medicine, 'innere Sicherheit' (internal security) in politics, or 'innere Angelegenheiten' (internal affairs).
Die Polizei ist für die innere Sicherheit zuständig.
The police are responsible for internal security.
outside
'außen' and 'inner' are clear opposites when describing position. 'außen' describes the outside of something. 'inner' describes something located on the inside, for example 'der innere Bereich' (the inner area) or 'die innere Wand' (the inner wall). They can be confused because both have to do with inside and outside. But the grammar is also important: 'außen' is an adverb and often stands on its own. 'inner' is an adjective and normally stands before a noun or appears with an ending. So you say 'außen ist es kalt' (it is cold outside), but 'der innere Teil' (the inner part).

Der innere Bereich des Gebäudes bleibt heute geschlossen.
The inner area of the building remains closed today.
Die Tür ist außen grün gestrichen.
The door is painted green on the outside.