Is the quotient of a regular local ring by a prime ideal Cohen-Macaulay? If so, how can we see this, if not, is there a counterexample? We know that a regular local ring is a UFD, so $0$ is a prime ideal, so in this case, the quotient is a regular local ring and hence a CM ring.
Any complete local noetherian ring is a quotient of a regular local ring. This is called Cohen's structure theorem for noetherian complete local ring (see for instance Matsumura Commutative Ring Theory Theorem 29.4). So any complete noetherian integral domain is a quotient of a local regular ring by a prime ideal. To construct a counterexample to your question, it is thus enough to exhibit a complete local noetherian domain which is not Cohen-Macaulay. The ring $k[X^4,X^3Y,XY^3,Y^4]$ (which is a non-CM domain) is usually a good starting point.
In the positive direction, your comment can be generalized to taking the quotient by any regular sequence (not necessarily a system of parameters).This gives a complete intersection ring, hence a CM ring.