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.

Well $2\cdot 3\equiv 1\; \text{mod}\ 5$, so $2$ and $3$ are multiplicative inverses $\text{ mod } 5$.
How to find the inverse of a number modulo a prime number was the topic of one of my previous answers.  Modulo a composite number, inverses don't always exist.
See Calculating the Modular Multiplicative Inverse without all those strange looking symbols for the way to find the inverse of $322$ mod $701$.  It turns out to be $455$
