$\mathbf{0}$ is the empty category, $\mathbf{1}$ is the category with one object, $*$, and one morphism, $1_*$, $\mathbf{2}$ is the category with two objects, etc. ...
Is there some system to these numbered categories? Could I come up with, say, category $\mathbf{15}$?
There are actually two different systems of "numbered" categories I know of. In one of them, if $n\in\mathbb{N}$, then $\mathbf{n}$ denotes a category with $n$ objects and no morphisms except the identity morphisms.
In the other, $\mathbf{n}$ denotes the poset $\{0,1,\dots,n-1\}$ with its usual order. That is, there are $n$ objects, one for each natural number $i<n$. For $i,j<n$, there is exactly one morphism $i\to j$ if $i\leq j$, and otherwise there are no morphisms $i\to j$.
For $n=0,1$ these definitions agree but for $n\geq 2$ they are different. In practice there is usually little risk of confusion, though, as it is easy to mention briefly what you mean and in any case it is usually easy to tell from context.