There are two monoids of order $2$ up to isomorphism:
\begin{array}{|c|c|c|} \hline & e & a \\ \hline e & e & a \\ \hline a & a & e \\ \hline \end{array} and \begin{array}{|c|c|c|} \hline & e & a \\ \hline e & e & a \\ \hline a & a & a \\ \hline \end{array}
The first monoid happens to be a group, namely the cyclic group $C_2$. Is there a "nice" or "standard" name for the other monoid in literature? Or a simple way to write down the monoid, in the same way that we may write $C_2$ as $\langle a: a^2 = e \rangle$. Note $(\mathbb{Z}/2\mathbb{Z}, \times)$ is isomorphic to this monoid, would this be a good candidate for such a name?
Similarly there are $7, 35, 228, \dots$ monoids of order $3, 4, 5, \dots$ (see this OEIS sequence). For those which are not groups, is there a nice way to refer to them?
The second one is denoted by $U_1$ in the literature on semigroups.
For the second part of your question, most finite semigroups are nilpotent (they have only one idempotent, which is a zero) and usually don't have a specific name. But there are specific names and notation for some subclasses of monoids or semigroups, like the Brandt semigroups, for instance.