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 M can be described as a direct sum where each summand S_i is of the form eta_i times R, where R is a polynomial rings (wose 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 have has remarkable combinatorial consequences.
