From my book, the definition given is:
Given a set $X$, a function $d: X \times X \to \mathbb{R}$ is a metric on $X$ if for all $x,y \in X \dots$
Then a metric space is a set $X$ together with a metric $d$.
What exactly is meant by the $X \times X$ here?
Is it essentially a cross product between any two components of $X$, which can be any sort of space, e.g complex plane, $\mathbb{R}^2$ etc ?
It is the Cartesian product defined by $X \times Y = \{ (x,y) | x \in X \quad y \in Y \} $
So for example $\{ 1 , 2 \} \times \{3,4\} = \{(1,3),(1,4),(2,3),(2,4)\}$