Question

what does monogamous nuclear family mean

Answer 1

 Monogamous marriage refers to one husband having one wife. (In polygamy, there can be multiple spouses. If there is more than one wife, it's called polygyny. If there's more than one husband, it's called polyandry.) 

Nuclear family is a term that refers to just two generations of "blood relatives." 

So, a monogamous, nuclear family would be mother, father, and their immediate children, with no other relatives or other people living as part of the defined family unit.