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

1. The group of units of $eMe$ is abelian for all idempotents $e$.

2. If $e,f$ are idempotents and $MeM=MfM$ the $ef$ is idempotent and $MefM=MeM$.

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 (http://arxiv.org/pdf/math/0702400.pdf). 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. You can also take the path semigroup of an acyclic quiver.

The theorem is that a finite monoid $M$ has only 1-dimensional irreps iff

1. The group of units of $eMe$ is abelian for all idempotents $e$.

2. If $e,f$ are idempotents and $MeM=MfM$ then $ef$ is idempotent and $MefM=MeM$.