Multiplicative group of integers modulo n definition issues 
It is easy to verify that the set $(\mathbb{Z}/n\mathbb{Z})^\times$ is closed under multiplication in the sense that $a, b ∈ (\mathbb{Z}/n\mathbb{Z})^\times$ implies $ab ∈ (\mathbb{Z}/n\mathbb{Z})^\times$, and is closed under inverses in the sense that $a ∈ (\mathbb{Z}/n\mathbb{Z})^\times$ implies $a^{-1} ∈ (\mathbb{Z}/n\mathbb{Z})^\times$.

The question is the following:
Firstly, are $a$ and $b$ referring to each equivalence class of integers modulo $n$?
Secondly, by $a^{-1}$, what is this referring to? If $a$ is the equivalence class, I cannot see (or I am not sure) how I can make inverse set.

First: yes, $a$ and $b$ represent the equivalence classes modulo $n$.
Second: Well, $a^{-1}$ is not the inverse as a rational number. Rather, it is the unique equivalence class modulo $n$ (if it exists) such that $aa^{-1}$ is congruent to 1 modulo $n$. And for existence, this inverse exists if and only if $a$ is coprime to $n$.
