Difference between $Z^*_n$ and $Z_n$ When studying cyclic groups I stumbled upon the following sentence:

The ∗ in $Z_n^*$ stresses that we are only considering mulitplication and forgetting about addition on this site of stanford. 

I don't really understand this definition, as I was learned that $Z_n^*$ just consists out of all elements of the elements of a cyclic group, minus the 0. Where does this the multiplication and addition come in?
$\mathbb{Z}_n^*$ doesn't mean $\mathbb{Z}_n - \{0\}$. You must remove all elements that are not invertible mod $n$, which is equivalent to keeping only the elements that are coprimes to $n$.

So, $\mathbb{Z}_n^* = \{x \in \mathbb{Z}_n : \exists x^{-1} \in \mathbb{Z}_n \} = \{x \in \mathbb{Z}_n : gcd(x, n) = 1\}$.

This is necessary since all elements in a group must have inverse elements (regarding the group operation). In $\mathbb{Z}_n$, the group operation is $+$, so, any $x$ has $-x$ as inverse mod $n$. But in $\mathbb{Z}_n^*$, the operation is the multiplication, so, some elements might not have inverses mod $n$.

For instance, take $\mathbb{Z}_9$. The elements $0, 3, $ and $6$ are not invertible mod 9 (we can't multiply them by other element and reduce mod $9$ to obtain $1$). Therefore, $\mathbb{Z}_9^* = \{1, 2, 4, 5, 7\}$.