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, which are introduced by Richard Brauer.
2.Temperley-lieb Algebras, which are quotients of Hecke algebras.
3.Partition Algebra, which are invented and studied by Paul Martin.
Both of the above examples are part of a wider branch which is called Diagram Algebras.