
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.

unique
Means 'einzigartig' (unique) or 'unvergleichlich' (incomparable) and is used to emphasize the special quality of something.
Die Atmosphäre bei diesem Konzert war einfach einzig.
The atmosphere at this concert was simply unique.
'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.
individual / separate
Einzig and einzeln can both refer to a particular thing, but they emphasize different aspects. Einzig means that nothing else exists; einzeln singles out one thing from a group without specifying the group’s total number. If there is only one error, einzig is the right term; when dealing with a particular error, einzeln is the right term.

Im Bericht wurde nur ein einziger Fehler gefunden.
Only one error was found in the report.
Im Bericht prüfen wir jeden Fehler einzeln.
In the report, we examine each error individually.