The following definitions are written in A primer on mapping class group.
$(1)$ An element $g$ of a group $G$ is primitive if there does not exist any $h\in G$ so that $g = h^k$ where $|k|>1$.
$(2)$ A closed curve in a surface $S$ (i.e. compact orientable $2$-dimensional manifold with possibly finitely many punctures or boundaries) is a multiple if it is a map $S^1\to S$ that factors through the map $S^1\xrightarrow{\times n}S^1$ for $n>1$.
Let $G = \pi_1(S)$. Then I thought not primitive = multiple and primitive = not multiple in $G$ because if $\gamma\in G$ is not primitive then there is $\alpha\in G$ such that $\gamma = \alpha^k$ for some $k>1$ (we can replace $\alpha$ to $\alpha^{-1}$) so $\gamma$ is multiple. If $\gamma$ is multiple then $\gamma = \beta^k$ for some $k>1$ and $\beta\in G$ so that $\gamma$ is not primitve.
But the book assumes there is not primitive and not a multiple element because there is a statement:
However, if $\alpha(\in\pi_1(S))$ is not primitive and not a multiple, then there are more lifts of $\alpha$ than there are conjugates. Indeed, if $\alpha = \beta^k$, then $\beta\langle\alpha\rangle\neq\langle\alpha\rangle$ while $\beta\alpha\beta^{-1} = \alpha$.
The first $\alpha =\beta^k$ is because we assume $\alpha$ is not primitive. I can't see where not multiple is used here. $\beta\langle\alpha\rangle = \langle\alpha\rangle\iff\beta\in\langle\alpha\rangle\iff\beta = \alpha^n$. This implies $\alpha$ has a finite order which is not ($\alpha$ is either hyperbolic or parabolic and clearly not identity).
The context of the above statement is the following:
When $S$ admits a hyperbolic metric and $\alpha$ is a primitive element of $\pi_1(S)$, we have a bijective correspondence: $$\left\{\text{Elements of the conjugacy class of }\alpha\text{ in }\pi_1(S)\right\}\leftrightarrow\left\{\text{Lifts to }\tilde{S} \text{of the closed curve }\alpha\right\}.$$ More precisely, the lift of the curve $\alpha$ given by the coset $\gamma\langle\alpha\rangle$ corresponds to the element $\gamma\alpha\gamma^{-1}$ of the conjugacy class $[\alpha]$.
If $\alpha$ is any multiple, then we still have a bijective correspondence between elements of the conjugacy class of $\alpha$ and the lifts of $\alpha$. However...