Arithmetically Cohen-Macaulay means that the section ring (with respect to a given ample line bundle) of the variety is Cohen-Macaulay.  It doesn't imply anything about Gorenstein-ness.   In fact, any projective variety with $H^i(X, \mathcal{O}_X) = 0$ for $0 < i < \dim X$ is arithmetically Cohen-Macaulay with respect to some embedding into projective space.  If I recall correctly, these notions appear most prominently in the study of Linkage (see Eisenbud's book for an introduction).  

A related notion is that of arithmetic Macaulayfication of a ring.  This means that there exists an ideal $I$ such that the Rees algebra of $I$ (the ring you blow-up to get the blow-up of $I$) is Cohen-Macaulay.  These were shown to exist in the last decade by Kawasaki.  If I recall correctly, a corollary of this result is that every ring with a dualizing complex is a quotient of a Gorenstein ring (this was previously a conjecture of Sharp).  Someone correct me if I'm wrong on this.