Dummit & Foote, $\langle g\rangle \leq C_G(g)$ for any $g \in G$. Let $G$ be a group and $C_G(g)$ denote the conjugacy class of $g$. In chapter 4.3 of Dummit and Foote, it is noted often that $\langle g\rangle \leq C_G(g)$ and this fact is used throughout some of the exercises (as a tool/hint) in the text. 
The following is an example (chapter 4.3 page 124)



But $C_{Q_8}(i) = \{\pm i \}$, it is not possible to reach $\pm 1$ in the conjugation $gig^{-1}$.



In this example $C_G(\cdot)$ is the centralizer (page 59 chapter 2.3)
Can someone clarify the first example for me?
 A: You are simply mistaken; $C_G(g)$ is denoting the centralizer of $g$ in $G$ throughout:
$$C_G(g) = \{x\in G\mid xg=gx\}.$$
The reason this subgroup shows up in the consideration of conjugacy classes is that there is a bijection between the cosets of $C_G(g)$ in $G$ and the conjugates of $g$ in $G$.
Indeed, if $xC_G(g) = yC_G(g)$, then there exists $h\in C_G(g)$ such that $x=yh$. Then 
$$\begin{align}xgx^{-1} & =(yh)g(yh)^{-1}\\ 
&= yhgh^{-1}y^{-1}\\ 
& = yghh^{-1}y^{-1} &\text{(since }hg=gh\text{)}\\ 
&= ygy^{-1}. 
\end{align}$$
And if $zgz^{-1}=wgw^{-1}$, then $w^{-1}zg = gw^{-1}z$, hence $w^{-1}z\in C_G(g)$, so $zC_G(g) = wC_G(g)$.
Thus, the size of the conjugacy class of $g$ is the index of $C_G(g)$ in $G$.
If we let $C_g = \{xgx^{-1}\mid x\in G\}$ be the conjugacy class of $g$, this amounts to
$$|C_g| = [G:C_G(g)]$$
which is a special case of the Orbit-Stabilizer Theorem. 
And of course, every power of $g$ commutes with $g$, so $\langle g\rangle \subseteq C_G(g)$ for all $g\in G$. 
A: Given a group $G$ and a subset $S\subseteq G$, then the centralizer $C_G(S)$ is the set of elements of $G$ that fix each element of $S$ under conjugation.  Equivalently, these are the elements of $G$ that commute with each of the elements of $S$.
So, for instance, if $G=Q_8$ and $S=\{i\}$, then you can check that the elements of $G$ that commute with $i$ are exactly $\{\pm 1,\pm i\}$.
On the other hand, the conjugacy class of $i$---the set of elements which are conjugate to $i$---is exactly $\{\pm i\}$.  It appears that you have confused $C_G(x)$ for the conjugacy class of $x$.
