In the previous answers almost every one 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.[Temperley-lieb Algebras](http://en.wikipedia.org/wiki/Temperley%E2%80%93Lieb_algebra), which are quotients of Hecke algebras. 2.[Partition Algebras](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 brach which is called Diagram Algebras.