What do we mean by a variety being arithmetically CohenMacaulay? Is every such variety also Gorenstein?

Arithmetically CohenMacaulay means, depending on the source/context, either:
Of course if you are projectively normal in $\mathbb{P}^n$ and the ample line bundle is a very ample line bundle of that embedding, these two definitions coincide. It doesn't imply anything about Gorensteinness. In fact, any CohenMacaulay projective variety with $H^i(X, \mathcal{O}_X) = 0$ for $0 < i < \dim X$ is arithmetically CohenMacaulay with respect to some embedding into projective space. To see this, take a sufficiently ample line bundle $L$ such that $H^i(X, \omega_X \otimes L^n) = 0$ and $H^i(X, L^n) = 0$ for all $n \geq 1$ and all $0 < i < \dim X$. In the previous version of this answer, I forgot the CohenMacaulay hypothesis on $X$, in which case the first vanishing can't be forced to hold. If I recall correctly, these notions appear 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 blowup to get the blowup of $I$) is CohenMacaulay. 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. EDIT: Added the CM hypothesis on the variety and added an explanation (thanks to Long). EDIT2: Added the two possible definitions (section ring vs coordinate ring). Thanks to J. C. Ottem. 

