With all of the genders available today, it's kind of amazing that German still has only three to choose from: feminine, masculine, and neuter (respectively die, der, and das in the nominative case). In German, there are two different ways to say "one of the" followed by a plural noun, such as "one of the cats," one of the dogs," or "one of the guinea pigs."

What makes this expression a bit difficult in German is that the gender in the "one" (in "one of") has to correspond to the gender of the plural noun you are using. This is a typical example of needing to know in German exactly what you're going to say before you say the first word of the sentence.

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Imagine you're at an animal shelter trying to decide between adopting a cat or a dog or a guinea pig, and are looking at one pen full of kitties, next to one full of puppies, and another full of guinea pigs. When they ask what kind of pet you are looking for, in English you could start saying "One of the..." and then finish the sentence once you'd decided. But not in German, because the "one" (einer/eine/eines) in German has to agree with the gender of animal you're about to pick: "cat" is die Katze, a feminine noun; "dog" is der Hund, a masculine noun; and "guinea pig" is das Meerschweinchen, a neuter noun.  So you'd have to make your decision before you open your mouth to say either:

Eine von den Katzen, bitte. (dative)
Eine der Katzen, bitte. (genitive)
One of the cats, please.

Einen von den Hunden, bitte. (dative)
Einen der Hunde, bitte. (genitive)
One of the dogs, please.

Eines von den Meerschweinchen, bitte. (dative)
Eines der Meerschweinchen, bitte. (genitive)
One of the guinea pigs, please.

Note that in the accusative case, the masculine einer becomes einen. As you can see, there is both a dative and a genitive way to say this. In spoken German, using the dative case is most common, whereas in formal written German, it's better to use the genitive. It's particularly tricky in that you need to know the gender of what you're discussing beforehand, and of course the definite articles are different in dative and genitive. But at least the plural definite article in dative is always den, and the plural definite article in genitive is always der.

Eines der letzten Spiele war dann von 1906 gegen 1860.

One of the last games was 1906 versus 1860.

Captions 15-16, Ball des Weines: Franz Beckenbauer

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The following example uses the adjective viel instead of the definite article:

Einer von vielen Punkten, die auch zwischen den Verhandlungspartnern hoch umstritten sind.

One of many points that are highly disputed between the negotiating parties.

Caption 34, Deutsche-Welle-Nachrichten: Massenprotest gegen TTIP

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When you use plural pronouns such as uns (us) or ihnen ("them") in a gender non-specific way, you always use the masculine form einer:

Einer von ihnen ist jüdisch, er geht jetzt hier zur Armee und so.

One of them is Jewish, he's going into the army here now and such.

Caption 28, Konstantin: ein Freiwilliger in Israel

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But if you are talking about women (die Frau, plural die Frauen) or children (das Kind, plural die Kinder), the genders must agree:

Es gibt 20 Frauen. Eine von ihnen...
Es gibt 20 Kinder. Eines von ihnen

Further Learning
Make a list of 20 or so nouns including their definite articles, such as die Frau, der Mann, das Kind. Then go through the list include each noun in a "one of the..." sentences, being sure to get the plural right. Don't forget too that many plural nouns in dative get an extra -n at the end: Einer von den Männern. Then go to Yabla German and watch the full videos above to see the context in which they have been used.