Look at these two sentences in English:
J'ai d'autres options.I have other options.
Je me fiche des autres options.I don't care about the other options.
When you need to express other [things] in French, you will use either d'autres or des autres.
Here's how to know when to use which !
1- When it is the plural of another [thing] :
(some) other [things] = d'autres [choses]
The indefinite adjective [un/une] autre - another + [singular noun] - becomes d'autres in the plural form.
-> Here the partitive des follows the rule of becoming de or d' in front of an adjective (See Using ''de / d' '' instead of 'des' in front of adjectives preceding nouns (partitive article)), hence d'autres [choses] and not des autres [choses].
Est-ce que vous avez d'autres livres?Do you have any other books?
J'ai eu une autre idée, oui, j'ai d'autres idées.I had another idea, yes, I have other ideas.
Je voudrais essayer un autre pull et d'autres pantalons.I'd like to try another jumper and other pairs of trousers.
ATTENTION:
You will never use autres on its own:
-> J'ai autres chaussures.
You can never use des autres to express some other [things]:
-> J'ai des autres chaussures.
2- When "other [things]" is used in expressions involving de (se servir de = to use; avoir besoin de = to need; avoir envie de = to feel like, etc)
Here look at these two examples:
Il a parlé de la France et d'autres pays.He spoke about France and about other countries.
Il a parlé de la France et des autres pays.He spoke about France and about the other countries.
des autres is the contraction here of [de + les autres] (about/of the others), so it implies that we're talking about specific things here (the others),
whereas using d'autres here is the contraction of [de + autres] (about/of others) is less specific ([some] others).
Here are more examples:
Il a besoin des autres feuilles de papier.He needs the other sheets of paper.
Il a besoin d'autres feuilles de papier.He needs other sheets of paper.
Elle se sert d'autres outils: ils sont plus légers.She uses other tools: they're lighter.
Elle se sert des autres outils: ils sont plus légers.She uses the other tools: they're lighter. -> Here the structure to say
to use [something] is
se servir de [quelque chose].
See also L'autre, les autres = The other one(s) (indefinite pronouns)
Elle se sert des autres outils: ils sont plus légers.She uses the other tools: they're lighter. J'ai eu une autre idée, oui, j'ai d'autres idées.I had another idea, yes, I have other ideas. Il a parlé de la France et d'autres pays.He spoke about France and about other countries. Je voudrais essayer un autre pull et d'autres pantalons.I'd like to try another jumper and other pairs of trousers. J'ai d'autres options.I have other options. Est-ce que vous avez d'autres livres?Do you have any other books? Je me fiche des autres options.I don't care about the other options. Il a besoin d'autres feuilles de papier.He needs other sheets of paper. Elle se sert d'autres outils: ils sont plus légers.She uses other tools: they're lighter. Il a besoin des autres feuilles de papier.He needs the other sheets of paper. Attention
Il a parlé de la France et des autres pays.He spoke about France and about the other countries. 
