How many different languages over the unary alphabet {a} are recognized by 2-state DFAs?
There are many ways to perform this enumeration. Note first that you can count the number of cases in half by postulating that $Q = \{q_0,q_1\}$, that is, fixing which among the two states is initial; the choice doesn't matter due to symmetry. Now you are left with 16 possibilities. At this point there are many ways to cut the search space even further. Here is one of them.

Case 1: $\delta(q_0,a) = q_0$. In this case we don't care about $\delta(q_1,a)$, since $q_1$ is unreachable. This case gives the two languages $\emptyset$ and $a^*$, depending on whether $q_0$ is a final state or not.

Case 2: $\delta(q_0,a) = q_1$. Since $F = \emptyset$ and $F = \{q_0,q_1\}$ will give the languages $\emptyset$ and $a^*$, respectively, it suffices to consider $F = \{q_0\}$ and $F = \{q_1\}$. Together, there are 4 machines to consider:


$\delta(q_1,a) = q_1$, $F = \{q_0\}$: the language is $\epsilon$.
$\delta(q_1,a) = q_1$, $F = \{q_1\}$: the language is $a^+$.
$\delta(q_1,a) = q_0$, $F = \{q_0\}$: the language is $(aa)^*$.
$\delta(q_1,a) = q_0$, $F = \{q_1\}$: the language is $a(aa)^*$.


These are the six different languages produced by unary two-state DFAs.