For me personally, the whole theory started to take shape (and make sense) once I learned about the graded case and understood connections with combinatorics.

For a graded (sometimes called $*$-local) ring, a basic technique for establishing the Cohen-Macaulay property is to deform it to a multigraded ring (=an affine semigroup ring) by exhibiting a SAGBI basis. For multigraded rings the question may often be decided by combinatorial techniques. More precisely, if $R_t$ is a flat deformation with a CM special fiber $R_0$ and general fiber $R$ then $R$ is also CM. A quotient $k[x_1,\ldots,x_n]/I$ of a polynomial ring by a square-free monomial ideal is a Stanley-Reisner ring of a simplicial complex $\Delta$ and CM property of the ring can be decided at the level of homology of $\Delta$ by the Reisner criterion. The corresponding simplicial complexes $\Delta$ are also called Cohen-Macaulay and have been much studied by people in algebraic combinatorics.

The Cohen-Macaulayness of determinantal rings mentioned in Hailong's answer can be established using the strategy I outlined (I think that Bruns and Herzog actually do it in a later chapter; can't verify since I don't have the book). "Combinatorial commutative algebra" by Miller and Sturmfels is well worth looking at for a more encompassing view. Stanley's "Combinatorics and commutative algebra" is older, but retains much of its appeal: it is very explicit and to the point. You can find many examples there.

For me personally, the whole theory started to take shape (and make sense) once I learned about the graded case and understood connections with combinatorics.

For a graded (sometimes called $*$-local) ring, a basic technique for establishing the Cohen-Macaulay property is "Gröbner degeneration": using a Gröbner basis, deform the ring to a quotient of a polynomial ring by a monomial ideal. Another approach is to deform a ring to a multigraded ring (=an affine semigroup ring) by exhibiting a SAGBI basis. This is known as "toric degeneration". The question then may be decided by combinatorial techniques. The commutative algebra bit is that if $R_t$ is a flat deformation with a CM special fiber $R_0$ and general fiber $R$ then $R$ is also CM. 

A quotient $k[x_1,\ldots,x_n]/I$ of a polynomial ring by a square-free monomial ideal is a Stanley-Reisner ring of a simplicial complex $\Delta$ and CM property of the ring can be decided at the level of homology of $\Delta$ by the Reisner criterion. The corresponding simplicial complexes $\Delta$ are also called Cohen-Macaulay and have been much studied by people in algebraic combinatorics.

The Cohen-Macaulayness of determinantal rings mentioned in Hailong's answer can be established using the strategy I outlined (I think that Bruns and Herzog actually do it in a later chapter; I can't verify it since I don't have the book). "Combinatorial commutative algebra" by Miller and Sturmfels is well worth looking at for a more encompassing view. Stanley's "Combinatorics and commutative algebra" is older, but retains much of its appeal: it is very explicit and to the point. You can find many examples there.