"les chats des enfants" simply means "the cats of the children", like "the top of the mountain" (le sommet de la montagne) or "the back of my phone" (le dos de mon téléphone), i.e. a noun + its complement attached with preposition: de/of. In French, you do not have a possessive form with <apostrophe S>, like "the children"s cat". you always use "de", for humans as for animals or objects.
Please note that in "les chats des enfants" "des" is in fact a contraction of "de-les" ; in "les chats du garçon", "du" is a contraction of "de-le".
That is a bit of a help, but it still does not tell me when 'des' indicates a possessive and when 'des' indicates 'some'. For example I get a sentence with 'des chiennes' and it is translated as 'some female dogs' and a few sentences later I get 'des chiennes' again and now it is a possessive form. What is the difference? Is there some cue that I am missing?
To have a possessive case, you need that something belongs to someone (human or animal) or something. Maybe you can find hereunder what you are looking for:
"Mary's dog is alone" = le chien de Mary est seul
"Mary's dogs are alone" = les chiens de Mary sont seuls
"the dog's nose is black" = le nez du chien est noir (du = contraction of de-le)
"dogs' eyes are generally black" = les yeux des chiens sont généralement noirs (des = contraction of de-les).
"some dogs are barking" = des chiens aboient (a number of them) "the dogs are barking" = les chiens aboient (these specific dogs) "dogs bark" = les chiens aboient (generality)
"the corner of the street" = le coin de la rue "the corners of the streets" = les coins des rues (contraction of de-les)