
annehmen
to accept
To accept or receive something that is offered to you.
Ich kann dieses teure Geschenk nicht annehmen.
I can't accept this expensive gift.
to accept / assume
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to accept
To accept or receive something that is offered to you.
Ich kann dieses teure Geschenk nicht annehmen.
I can't accept this expensive gift.

to assume
To suppose something or consider it likely without knowing it for certain.
Ich nehme an, der Zug hat Verspätung.
I assume the train is delayed.

to adopt
To officially approve and pass a proposal, law, or motion by vote.
Der Gesetzesentwurf wurde vom Parlament einstimmig angenommen.
The draft law was adopted unanimously by parliament.

to take care of
To take care of a person, problem, or task and assume responsibility for it.
Die Nachbarin hat sich des verwaisten Kätzchens angenommen.
The neighbor took care of the orphaned kitten.

to take on
To gradually acquire a particular quality, form, or state.
Die Blätter der Bäume nehmen im Herbst eine rote Farbe an.
The leaves of the trees take on a red color in autumn.
'Annehmen' (to accept / assume) is a separable verb. The prefix 'an-' goes at the end in a main clause, stays attached to the verb in a subordinate clause, and in the infinitive with 'zu', 'zu' goes in between.
Er nimmt das Paket für seine Nachbarin an.
He accepts the package for his neighbor.
If you want to express an assumption, the structure 'Ich nehme an, dass ...' (I assume that ...) is very common. The subordinate clause with 'dass' then contains the assumption.
Ich nehme an, dass die Besprechung um 10 Uhr beginnt.
I assume that the meeting starts at 10 o'clock.
The expression 'sich einer Sache annehmen' (to take care of something) requires the genitive. This is a fixed structure that occurs mainly in written language.
Der Bürgermeister will sich des Problems der alten Schule annehmen.
The mayor wants to take care of the problem of the old school.
In the meaning 'akzeptieren' (to accept), 'annehmen' (to accept) is often combined with things that you are offered, for example: to accept an offer, an invitation, help, or an apology.
Er hat ihre Entschuldigung sofort angenommen.
He accepted her apology immediately.
This meaning of 'annehmen' (to adopt) is used almost exclusively in formal, political, or legal contexts. Typical signal words are 'Parlament' (parliament), 'Gesetz' (law), 'Antrag' (motion/application), or 'Vorschlag' (proposal).
Der Antrag wurde mit großer Mehrheit angenommen.
The motion was adopted by a large majority.
When something acquires a new quality, 'annehmen' (to take on) is often combined with words like 'Farbe' (color), 'Form' (form), or 'Gestalt' (shape).
Der Teig muss eine feste Form annehmen, bevor er gebacken wird.
The dough must take on a firm shape before it is baked.
to record / take in
For a new member, annehmen refers to approving the application. Aufnehmen refers to the person subsequently joining the group. A club can therefore first accept the application and then admit the person as a member.

Der Vorstand nimmt Leas Antrag auf Mitgliedschaft an.
The board accepts Lea's application for membership.
Der Verein nimmt Lea als neues Mitglied auf.
The club admits Lea as a new member.