One way I think about Cohen-Macaulayness (probably not in largest generality but at least in context relevant to combinatorics) is as follows:

Think first of the ring of symmetric polynomials in $n$ variables. A remarkable fact from first year linear algebra is that this ring is a polynomial ring in some other variables, the elementary symmetric polynomials. 

Being a polynomial ring is rare (but this is a sort of role model). Being Cohen-Macaulay comes close. A Cohen Macaulay ring can be described as a direct sum where each summand $S_i$ is of the form $\eta_i R$, where $R$ is a polynomial rings (whose variables are the elements of a system of parameters) and $\eta_i$ are elements. Being a direct sum is important here. 

For graded rings such a description has remarkable combinatorial consequences.