This is the 194th most frequent Ukrainian word.
нашої
"Нашої" means "of our" or "ours" (genitive singular feminine form).
'Нашої' is used here to denote possession in the genitive case, referring to 'сім'ї' (family, which is feminine singular). This shows that the picture belongs to 'our' family.
Це картина нашої сім'ї.
This is a painting of our family.
Here, 'Нашої' describes 'вчительки' (teacher, feminine singular, genitive), again showing belonging or association.
Це скрипка нашої вчительки музики.
This is our music teacher's violin.
In this instance, 'Нашої' modifies 'однокласниці' (classmate, feminine singular, genitive), indicating possession.