einzig
only
Describes that there is only one instance of something or someone and no other.
Das ist der einzige Schlüssel, den wir für diese Tür haben.
That is the only key we have for this door.

only / unique
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only
Describes that there is only one instance of something or someone and no other.
Das ist der einzige Schlüssel, den wir für diese Tür haben.
That is the only key we have for this door.

'Einzig' is derived directly from the number 'eins' (one). It emphasizes that something exists only once or stands alone in its kind.
Er war der einzige Gast auf der Party.
He was the only guest at the party.
The construction 'der/die/das Einzige, was/der/die...' (the only thing/person that/who...) is very common. It is used to set one thing or person apart from all others.
Das Einzige, was ich jetzt möchte, ist ein Glas Wasser.
The only thing I want right now is a glass of water.
In the meaning 'einzigartig' (unique), 'einzig' often appears at the end of the sentence (predicatively) and is not declined. It then functions like an adverb and strengthens a statement.
Die Aussicht vom Berggipfel war einfach einzig.
The view from the mountain peak was simply unique.
only
'nur' and 'einzig' both have to do with limiting something to one thing or person. 'nur' is an adverb and can be placed quite freely in a sentence: it limits a statement. 'einzig' is usually an adjective and describes that there really is only one instance or one possibility. 'einzig' sounds stronger and more precise when you want to emphasize that there is no other person, thing, or solution. They can be confused because both can express 'not more than one'. In practice: 'nur' limits the statement, while 'einzig' describes the thing itself.

Meine Freundin hat nach der Arbeit die einzige freie Stunde zum Lernen genutzt.
After work, my friend used her only free hour to study.
Er hat am Montag nur eine Aufgabe für die Schule bekommen.
On Monday, he only got one assignment for school.