Let $A \subseteq \chi$ and $\epsilon > 0$. We say that A is $\epsilon$-separated if $\rho(x;y) \geq \epsilon$ for all $x,y \in A$, where $\rho$ is a Metric over the set $\chi$.

The $\epsilon$-packing number of $B \subseteq X$, denoted by $\mathcal{M}(\epsilon;B)$ is the size of the largest $\epsilon$-separated set contained in $B$.

It is the same as the size of the largest number of centers of balls of size $\epsilon/2$ we can pack inside the set $B$, hence why it's called the packing number.

Note that if a point in $B$ is further from all points in $A$ than $\epsilon$, we can always add one more point to $A$, therefore the maximal packing set (of size $\mathcal{M}(\epsilon;B)$), has all points in $B$ within less than distance $\epsilon$ of points in $A$, and it is therefore a covering set. Therefore, $\mathcal{M}(\epsilon;B)$ is an upper bound on the Covering number

See here and here and UML