Why Cohen-Macaulay rings have become important in commutative algebra? 
I want to know the historic reasons behind singling out Cohen-Macaulay rings as interesting algebraic objects. 

I'm reviewing my previous lecture notes about Cohen-Macaulay rings because now I'm studying about Stanley-Reisner rings and I think I need to have a better general understanding about why I need to study CM rings.
 A: Richard Stanley's description on How the Upper Bound Conjecture was Proved tells much about the history of how Cohen-Macaulay rings became important in combinatorics. Additional links can be found here.
A: I think apart from commutative algebra, one main importance of the Cohen-Macauley is in Geometry. CM rings lie between the regular rings and other types of (bad) singularities. In a sense CM singularities are the best singularities and are much easier to deal with. One of the simplest properties of CM singularities is for example that they are equidimensional. 
A: I think there are many reasons.  Here are a few.
Practical reasons
Cohen-Macaulay rings are just plain easier to work with.  
Computations in local cohomology
For example, any number of computations in local cohomology modules become much easier in the Cohen-Macaulay case (see for example Bruns and Herzog's book on the topic).  Explicitly, it's much easier to determine if a class in $z \in H^{\dim R}_{\mathfrak{m}}(R)$ is zero or not in the case that $R$ is Cohen-Macaulay.
Duality
Both Grothendieck-local and Grothendieck-Serre duality work much better in Cohen-Macaulay rings.  The dualizing complex (assuming it exists) is a complex whose first non-zero cohomology is the canonical module and which is equal to this (shifted) canonical module if and only if the ring is Cohen-Macaulay.  Without this hypothesis one frequently needs to work in the derived category and do numerous computations with spectral sequences.  It is convenient to not have to.
Vanishing and exactness
If $R$ is Cohen-Macaulay and $I$ is a height-one ideal (and suppose the rings are quotients of Gorenstein/regular rings so they have dualizing complexes).  Then we have a surjection of canonical modules $\text{Hom}_R(I, \omega_R) = \omega_R(I^{-1}) \to \omega_{R/I}$.  This is surjective because the next term is zero  when $R$ is Cohen-Macaulay. This sort of vanishing applies to more general situations and is really useful (there is a local dual version involving local cohomology).  There are lots of other vanishing results that you can deduce from this kind too.
Ubiquity of Cohen-Macaulay rings
Ok, if Cohen-Macaulay rings weren't so common, the above nice properties would be less interesting.  But Cohen-Macaulay rings are really common.  Here are some examples.
Summands of regular (or Cohen-Macaulay) rings
If $R \subseteq S$ is a extension of rings and $R \to S$ splits as a map of $R$-modules, then if $S$ is Cohen-Macaulay, so is $R$ (the point is $H^i_m(R) \to H^i_{mS}(S)$ injects and the latter term is zero, at least after a little localization on $S$ if necessary).  Lots of rings coming from representation theory for instance are summands of regular rings.  
Complete intersections
Complete intersection rings are Cohen-Macaulay.
Rational/log terminal/F-regular singularities  A lot of classes of singularities which are most useful today are Cohen-Macaulay.  One of their most useful properties is their vanishing properties (see above).

Pithy quotes
"Life is really worth living in a Noetherian ring $R$ when all the local rings
have the property that every s.o.p. is an $R$-sequence. Such a ring is called
Cohen-Macaulay (C-M for short)."
[Page 887 of Hochster, Some applications of the Frobenius in characteristic 0 ]
A: I'm no expert on the evolution of Cohen-Macaulay rings, so I will leave that part of your question for those who actually know their history. 
On a high level, Cohen-Macaulay rings are wonderful precisely because they lie within the intersection of algebraic geometry, algebraic topology, combinatorics, commutative algebra, and probably another four or five active research fields which I lack the perspective to mention. As such, there are plenty of "hooks" for people with diverse backgrounds to use what they know to prove something cool in a seemingly disparate field. I haven't touched commutative algebra with a ten-foot pole in my own research, but hey: I understand link conditions on simplicial complexes, so I have something to latch on to when I read about C-M rings.
For a precise sequence of as many as 9 answers to your specific question that were relevant as early as 1978, I would recommend the introduction of this paper by Melvin Hochster. It is called "Cohen Macaulay Rings and their Modules".

Update (Nov 24, 2014): A fairly comprehensive introduction to the importance of Cohen-Macaulay complexes can now also be found in Anders Björner's recent article dedicated to Richard Stanley on his 70th birthday.
