eins
one
The digit or number 1, used as a cardinal number when counting or calculating.
Wir zählen zusammen bis drei: eins, zwei, drei.
Let's count together to three: one, two, three.

one
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one
The digit or number 1, used as a cardinal number when counting or calculating.
Wir zählen zusammen bis drei: eins, zwei, drei.
Let's count together to three: one, two, three.

The form 'eins' with '-s' at the end is used when counting or when the number stands on its own, without a noun directly following it. It can then replace a neuter noun or be used as an abstract number.
Wir zählen jetzt: eins, zwei, drei.
We're counting now: one, two, three.
In the German school system, 'eine Eins' (an A / a 1) is the best grade ('sehr gut', very good). The expressions 'eine Eins schreiben' (to get an A / a 1 on a test) or 'eine Eins bekommen' (to get an A / a 1) are very common in everyday school language.
Er hat in der Matheprüfung eine Eins geschrieben.
He got an A on the math exam.
The expression 'sich eins sein' means 'sich einig sein' (to be in agreement) and is always used with the verb 'sein' (to be) and a reflexive form. The people who agree are in the plural.
In diesem Punkt sind wir uns zum Glück alle eins.
Fortunately, we all agree on this point.
In colloquial speech, 'eins' is often used as a pronoun for 'etwas' (something) or 'eine Sache' (one thing). It often comes at the beginning of a sentence to emphasize a particular thing.
Eins kann ich dir versprechen: Langweilig wird es nicht.
One thing I can promise you: it won't be boring.
In colloquial speech, 'eins' is often used for the time 'ein Uhr' (one o'clock). You can say both 'Es ist ein Uhr' (It is one o'clock) and 'Es ist eins' (It is one).
Wir treffen uns um eins am Bahnhof.
We're meeting at one at the train station.
a / one
'ein' and 'eins' both have to do with the number 1, but they appear in different places in a sentence. 'ein' comes before a masculine or neuter noun, for example 'ein Mann' (a man) or 'ein Kind' (a child). It can be an indefinite article or emphasize the quantity: exactly one. You use 'eins' when the number stands on its own, for example when counting or doing calculations. Learners easily confuse the words because both point to the number 1. A simple rule is: before a suitable noun, it is usually 'ein'; on its own as a number, it is 'eins'.

Die Kinder zählen laut: null, eins, zwei, drei.
The children count out loud: zero, one, two, three.
Vor dem Wochenende brauche ich noch ein Rezept vom Arzt.
Before the weekend, I still need a prescription from the doctor.