"Do you like our district?"
Translation:Тебе нравится наш район?
"наш" is the masculine accusative form of the first person plural possessive pronoun (i.e. "our"), and must be used here because "our district" is the object of the sentence, and is hence in the accusative. I'm not too sure what 'наший' is, do you mean "нашей"? If so, "наший" is the feminine genitive, dative, instrumental and prepositional first person plural possessive pronoun, and wouldn't be used here.
See this link (and Wikitionary in general; it's very helpful) for the full declension table of наш: https://en.wiktionary.org/wiki/%D0%BD%D0%B0%D1%88#Russian