Taking the attitude that a category is interesting not so much in its own right as for the functors into or out of it, I like the category with just two objects, called $A$ and $V$, and with just four morphisms, namely the two identity morphisms and two other morphisms, called $s$ and $t$, both pointing from $A$ to $V$. A functor $F$ from this category to the category of sets is "the same thing" as a directed graph (allowing loops and multiple edges). $F(V)$ is the set of vertices of the graph, $F(A)$ is the set of arrows, and $F(s)$ and $F(t)$ assign to each arrow its source and its target, respectively.

Continuing to take the same attitude, I also like any finite group as an example of a category with just one object; the morphisms are the elements of the group and composition is the group's operation. A functor from this to the category of sets is a permutation action of the group. A functor from the group to the category of vector spaces over a field $k$ is a $k$-linear representation of the group.