I am told to consider the set $M_2(\mathbb{Z})$ as a group under addition and the set $M_2(n\mathbb{Z})$ as a normal subgroup.
However I have no idea whether the second set is supposed to be $M_2(n+\mathbb{Z})$ or $M_2(n*\mathbb{Z})$
The full question is to show that $M_2(n\mathbb{Z})$ is isomorphic to $M_2(\mathbb{Z}_n)$
What is likely is that the question asks you to show that the quotient $M_2(\mathbb{Z})/M_2(n\mathbb{Z})$ is isomorphic to $M_2(\mathbb{Z}_n)$.
If this is the case, then $n\mathbb{Z}$ indicates multiplicative multiples of $n$: $$n\mathbb{Z}=\{\ldots, -n,0,n,2n,\ldots\}.$$ So, $M_2(n\mathbb{Z})$ is the subset of $2\times 2$ matrices such that each entry is a multiple of $n$.