
sich einigen
to agree
To come to a mutual agreement or compromise after a discussion or negotiation.
Die beiden Parteien konnten sich schließlich auf einen Preis einigen.
The two parties were finally able to agree on a price.
to agree / unite
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to agree
To come to a mutual agreement or compromise after a discussion or negotiation.
Die beiden Parteien konnten sich schließlich auf einen Preis einigen.
The two parties were finally able to agree on a price.

to unite
To bring different people, groups, or parties together into one unit; to unify.
Der Anführer versuchte, die zerstrittenen Stämme zu einigen.
The leader tried to unite the feuding tribes.
The most common use is reflexive: 'sich einigen' (to agree) means to reach an agreement. The rarer, non-reflexive form 'jemanden einigen' (to unite someone) means to bring different groups together.
Die Geschwister konnten sich nicht einigen, wer das Auto bekommt.
The siblings could not agree on who would get the car.
The verb 'einigen' (to agree/unite) comes from the number word 'eins' (one). The basic idea is to make 'one' unit or 'one' agreement out of several parts or opinions.
Nach langer Diskussion waren sie sich endlich einig.
After a long discussion, they finally agreed.
When you want to say what people agree on, use the preposition 'auf' followed by the accusative.
Die Koalitionspartner haben sich auf ein neues Gesetz geeinigt.
The coalition partners agreed on a new law.
to agree / reconcile
Both verbs refer to a joint agreement. With vereinbaren, the specific arrangement is usually the object: one vereinbart conditions. Sich einigen focuses on the people involved and the consensus they have reached: they sich einigen auf these conditions.

Die Parteien einigten sich auf eine Zahlungsfrist von drei Monaten.
The parties agreed on a payment period of three months.
Die Parteien vereinbarten eine Zahlungsfrist von drei Monaten.
The parties agreed on a payment period of three months.