äußer
outer
Describes a position that is farther outside or on the periphery of something, as opposed to the inside or center.
Wir wohnen im äußeren Ring der Stadt, wo es ruhiger ist.
We live in the outer ring of the city, where it is quieter.

outer / external
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outer
Describes a position that is farther outside or on the periphery of something, as opposed to the inside or center.
Wir wohnen im äußeren Ring der Stadt, wo es ruhiger ist.
We live in the outer ring of the city, where it is quieter.

The adjective 'äußer-' (outer/external) is a comparative form derived from the adverb 'außen' (outside). It describes something that is located farther outside than something else.
Die äußere Schale der Nuss ist sehr hart.
The outer shell of the nut is very hard.
The adjective 'äußer-' (outer/external) is almost always used directly before a noun and must be declined. It normally does not stand alone at the end of a sentence.
Der äußere Eindruck kann täuschen.
The outward impression can be deceptive.
In its spatial meaning, 'äußer-' (outer) is often combined with nouns that describe a boundary or area, e.g. 'der äußere Rand' (the outer edge), 'die äußere Mauer' (the outer wall), or 'der äußere Stadtteil' (the outer district).
An der äußeren Grenze des Grundstücks steht ein Zaun.
There is a fence at the outer boundary of the property.
In the figurative sense, 'äußer-' (external/outward) often refers to appearance. Typical combinations are 'die äußere Erscheinung' (outward appearance), 'der äußere Schein' (outward appearance), or 'äußere Umstände' (external circumstances).
Lass dich nicht vom äußeren Schein blenden.
Do not let yourself be blinded by outward appearances.
outside
'außen' and 'äußer' both have to do with the outside. 'außen' is an adverb and says where something is: not inside, but on the outside. 'äußer' is an adjective and stands before a noun, for example 'die äußere Tür' (the outer door) or 'der äußere Rand' (the outer edge). It describes a position farther outside or something visible from the outside. Learners may confuse the words because they look similar. The grammar is important: you say 'Das Schild hängt außen' (The sign is hanging outside / on the outside), but 'die äußere Wand' (the outer wall).

Am äußeren Rand des Bahnhofs stehen mehrere Taxis.
Several taxis are waiting at the outer edge of the train station.
Die Jacke ist innen warm und außen wasserfest.
The jacket is warm on the inside and waterproof on the outside.