The question is vague. I am assuming you want finite monoids over the complex field, although I could answer over any field, and I am assuming you want irreducible reps. There are noncommutative monoids whose irreducible representations are all 1-dim. I characterized with Almeida, Margolis and Volkov all such monoids. The simplest example is take the three element monoid consisting of $1,a,b$ where 1 is the identity and $xy=x$ if $x$ is not 1.
The theorem is that a finite monoid $M$ has only 1-dimensional irreps iff
The group of units of $eMe$ is abelian for all idempotents $e$.
If $e,f$ are idempotents and $MeM=MfM$ the $ef$ is idempotent and $MefM=MeM$.