In the previous answers, almost everyone  stressed on the reasons why have the representations of monoids not been studied widely, which probably answers your question.

But I would like to mention a couple interesting cases were the structure of monoid algebras are fully understood (over the field of complexes).

1.[Brauer Algebras](https://en.wikipedia.org/wiki/Brauer_algebra), which are introduced by  Richard Brauer.

2.[Temperley-lieb Algebras](http://en.wikipedia.org/wiki/Temperley%E2%80%93Lieb_algebra), which are quotients of Hecke algebras.

3.[Partition Algebra](http://www.worldscientific.com/doi/abs/10.1142/S0218216594000071), which are invented and studied by Paul Martin.

Both of the above examples are part of a wider branch which is called Diagram Algebras.