I read the following on this Wikipedia page:
Let ${\displaystyle \mathbb {Z} _{q}}$ denote the ring of integers modulo ${\displaystyle q}$ and let ${\displaystyle \mathbb {Z} _{q}^{n}}$ denote the set of ${\displaystyle n}$-vectors over${\displaystyle \mathbb {Z} _{q}}$.
What I don't get (being a noob) is what the members of ${\displaystyle \mathbb {Z} _{q}^{n}}$ might be like:
Say ${\displaystyle q}$ is $5$, then will the members of ${\displaystyle \mathbb {Z} _{q}^{n}}$ be of the form such that for every $x \in \displaystyle \mathbb {Z} _{q}^{n}$, there is a one-to-one correspondence to a member in $\displaystyle \mathbb {Z} _{q}$, for example a binary encoding of the integers $0, 1,... 4$? Or does it simply mean that the $n$-vector members are such that every element in the vector is in ${\displaystyle \mathbb {Z} _{q}}$.
Your last sentence is the right interpretation.
$$\mathbb{Z}_p^n = \mathbb{Z}_p \times \cdots \times \mathbb{Z}_p = \{(x_1 , \ldots, x_n) \, | \, x_1, \ldots x_n \in \mathbb{Z}_p \}$$
$\mathbb{Z}_p^n$ may be viewed as a ring with the usual addition and multiplication $\mod p$ defined componentwisely.
Example: Two elements in $\mathbb{Z}_5^3$ are $(1, 2, 3)$ and $(2, 4, 3)$. We have
$(1, 2, 3) + (2, 4, 3) = (3, 1, 1) \quad \text{and} \quad (1, 2, 3)(2, 4, 3) = (2, 3, 4)$