What does the notation $\mathbb{Z}[X]$ refer to? Suppose that a mathematician wrote,

Let $f_0, f_1, f_2, \cdots$ be functions in $\mathbb{Z}[X]$

What does $\mathbb{Z}[X]$ mean?
I am aware that $\mathbb{Z}$ is used to denote the set of integers.
In other words, $\mathbb{Z} = \{\cdots, -100000, \cdots, -3, -2, -1, 0, +1, +2, +3, \cdots, +100000, \cdots\}$
What on earth is $\mathbb{Z}[X]$?

$\mathbb Z[X]$ is the ring of polynomials with integer coefficients. Written out in set-builder notation: $\mathbb Z[X]=\{\sum_{i=0}^{n} a_iX^i|i\in\mathbb N,a_i\in\mathbb Z \}$. You can plug in values of $\mathbb Z$ in for $X$ to get back an element of the base ring $\mathbb Z$. Though some authors may refer to a specific polynomial as "$f(x)\in \mathbb Z$," we consider polynomials as distinct algebraic objects here rather than  functions – we are not considering the polynomial as a morphism/mapping between two groups/rings/fields in the same way we would think about a homomorphism. $X$ in this case is considered an indeterminate, not a variable (you can read more about this here).
$\mathbb Z[X]$ is a countably infinite ring (you can prove this by checking the ring axioms as an exercise!), but not a field – all polynomial rings are not closed under multiplicative inverses. This same notation applies to polynomial rings over other rings/fields. For example, $\mathbb Z_2[X]=\{\sum_{i=0}^{n} a_iX^i|i\in\mathbb N,a_i\in\mathbb Z_2 \}$ is the set of polynomials with coefficients $0$ or $1$. You can generate an ideal, a special subgroup of the polynomial ring, using one such $f(x)\in \mathbb Z$.
