"They do not have water."
Translation:У них нет воды.
Two different points of confusion:
(1) Why is "they" in the prepositional here, whereas "I have" is written "у меня," with I in the genitive case?
(2) Why is the thing being owned in the genitive case? Word-for-word I would parse this sentence as "Water is not by them," and would assume "water" is nominative.
1) "Них" is genitive, too. Have you been confused by Н? It is added to pronouns starting with vowels when they are used with prepositions. Их - у них, его - от него, её - без неё, etc.
2) You use genitive in negations of this kind. У [someone, genitive] нет [something, genitive].
Regarding your second point, is this a little like saying "[at _ is] none of __"? Where the of kinda suggests genitive perhaps?
(I'm not sure if the "at" is strictly the correct literal translation but I did some Irish before this and I think that's how their similar structure would be literally translated. Anyway, yes, I know it's equivalent to "to have")