Atomic, reduced monoids fit the bill here. If $H$ is a multiplicatively written monoid with identity $1_H$, then an *atom* of $H$ is an element $a \in H \setminus H^\times$ for which there do not exist $x, y \in H \setminus H^\times$ such that $a = xy$, where $H^\times$ is the group of units of $H$. In particular, $H$ is called *atomic* if every non-unit of $H$ is a (finite) product of atoms. On the other hand, we say that $H$ is *reduced* if $H^\times = \{1_H\}$.

There is a vast literature on the factorization theory of atomic monoids, though most of it is centered on the *commutative* and *cancellative* setting, for which you may want to have a look to 

> A. Geroldinger and F. Halter-Koch, *Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory*, Pure Appl. Math. **278**, Chapman & Hall/CRC, Boca Raton (FL), 2006.

If, on the other hand, you are also interested in non-cancellative or non-commutative monoids, then it's a totally different story, and the best I can do is to address you to my own work with Yushuang Fan, where you will find, if interested, some pointers to relevant literature (most notably, Smertnig's and Baeth and Smertnig's work on cancellative categories) and an entire section devoted to fundamental aspects of the theory (namely, Sect. 2):

> Y. Fan and S. Tringali, *Power monoids: A bridge between Factorization Theory and Arithmetic Combinatorics*, preprint ([arXiv:1701.09152][1]).

Lastly, if you find yourself wondering about sufficient criteria for a monoid to be atomic, then you can find something in [another thread][2] on this forum. 

Of course, atomic monoids are *not* the only semigroups that fit your requests. But they are certainly the only class for which a systematic theory of factorization has been so far developed.


  [1]: https://arxiv.org/abs/1701.09152
  [2]: https://mathoverflow.net/questions/264379/