The set of homotopy classes of maps from $X$ to $Y$ is denoted by $[[X, Y]]$. An element of $[[X, Y]]$ is denoted usually by $[[f]]$.
Given a map $\alpha : Y \to Z$, there is a function $\alpha_* : [[X, Y]] \rightarrow [[X,Z]]$ defined by post-composing with $\alpha$, viz., $f \rightarrow \alpha \circ f$.
Can someone explain an example of this? I can't see how the map $f \rightarrow \alpha \circ f$ maps between the two equivalence classes. How would you write $\alpha_{*}([[f]])$ with this composition relationship? Is it $[[\alpha \circ f]]$? This isn't explicitly explained in the book I'm reading.
You are correct, $\alpha_*[[f]] = [[\alpha\circ f]]$.
Let $\mathcal{C}(X, Y)$ denote the set of continuous functions $X \to Y$. There is a quotient map $\pi : \mathcal{C}(X, Y) \to [[X, Y]]$ given by $f \mapsto [[f]]$.
If $\alpha : Y \to Z$ is a continuous map, there is an induced map $\alpha_* : \mathcal{C}(X, Y) \to \mathcal{C}(X, Z)$ given by $f\mapsto \alpha\circ f$. If $f$ and $g$ are homotopic, then so are $\alpha\circ f$ and $\alpha\circ g$, i.e. if $[[f]] = [g]]$, then $[[\alpha\circ f]] = [[\alpha\circ g]]$. Therefore the map $\alpha_*$ descends to a well-defined map $[[X, Y]] \to [[X, Z]]$, again denoted by $\alpha_*$, given by $[[f]] \to [[\alpha\circ f]]$. This is represented in the commutative diagram below
$$\require{AMScd} \begin{CD} \mathcal{C}(X, Y) @>{\alpha_*}>> \mathcal{C}(X, Z)\\ @V{\pi}VV @VV{\pi}V \\ [[X, Y]] @>{\alpha_*}>> [[X,Z]]. \end{CD}$$