beide
both
Refers to two people or things and means the one and the other together.
Ich habe Anna und Tom gefragt, und beide haben zugestimmt.
I asked Anna and Tom, and both agreed.

both
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both
Refers to two people or things and means the one and the other together.
Ich habe Anna und Tom gefragt, und beide haben zugestimmt.
I asked Anna and Tom, and both agreed.

'Beide' can stand alone as a pronoun or directly before a plural noun as an article-like word. In both cases, it refers to exactly two people or things that are already known.
Ich habe zwei Brüder. Beide leben in Berlin.
I have two brothers. Both live in Berlin.
'Beide' is always plural and is declined like an adjective. The ending adapts to the case. For example: 'Ich kenne beide' (I know both of them; accusative), 'Ich spreche mit beiden' (I speak with both of them; dative), or 'Das ist die Meinung beider' (That is the opinion of both; genitive).
Der Lehrer hat beiden Schülern eine gute Note gegeben.
The teacher gave both students a good grade.
To express that neither of two options applies, you use 'keiner von beiden' or 'keine von beiden' (neither of the two). This expression negates the entire group of two.
Welches Kleid gefällt dir besser? Ehrlich gesagt, keines von beiden.
Which dress do you like better? Honestly, neither of them.
no / not any
'kein' and 'beide' can both refer to a choice, but they say opposite things. 'beide' means two people or things together: one and the other. 'kein' negates a noun and says that nothing or no one applies. With exactly two things, people often say 'keiner von beiden' (neither of the two / neither of them) when neither one nor the other fits. Learners may confuse the words because 'beide' and 'keiner von beiden' can occur in the same situation. The important point is: 'beide' means two together; 'keiner von beiden' means zero out of two.

Elif nimmt beide Bücher mit in den Urlaub.
Elif is taking both books with her on vacation.
Von den zwei Terminen passt mir leider keiner.
Unfortunately, neither of the two appointments works for me.