Some interesting examples of Cohen-Macaulay but not Gorenstein rings:
Maximal minors: Let $m\geq n$ be integers. Take $S=k[x_{ij}]$ with $1\leq i\leq m, 1\leq j\leq n$ and $I$ be the ideal generated by all $n$ by $n$ minors. Then $R=S/I$ is CM, but is Gorenstein if and only if $m=n$.
Veronese subrings: Let $S=k[x_1,\cdots,x_n]$ and $R=S^{(d)}$ be the $k$-subalgebra of $S$ generated by the monomials of degree $d$. Then $R$ is always CM, but is Gorenstein if only if $d$ devides $n$.