In AdS/CFT, what is on the AdS side, supergravity or string theory? What really is on the AdS side in AdS/CFT? Does it always have to be string theory or is sometimes supergravity "enough" or better suited to do calculations?
From the answers to my earlier question, I have learned that one can calculate CFT/QFT correlation functions on the boundary from the quantum gravitational partition function valid inside the AdS space by taking the boundary value
$$
\langle O(x_1)O(x_2)\cdots O(x_n)\rangle \sim \frac{\partial^n Z}{\partial \Phi_0(X_1)\partial \Phi_0(X_2)\cdots\partial \Phi_0(X_n)}
$$
Does the action that appears in the partition function on the AdS side, that is
$$
Z = e^{−S(\Phi)}
$$
have to come from supergravity or string theory?
When reading about AdS/CFT I have seen it defined with both possibilities and this confuses me.
So when and why does it make a difference, of one assumes strings or supergravity to calculate the partition function on the AdS side? Are there cases when one or the other is more appropriate, simpler, useful, etc?
 A: Since 1998 or earlier, there have been no doubts that the AdS/CFT correspondence provides us with a full non-perturbative definition of string theory on the AdS-like background, including all of (type IIB) stringy objects and interactions and subtleties that we have ever heard of. An obvious reason why the CFT can't be equivalent "just to supergravity" is that the pure supergravity is inconsistent as a quantum theory while the CFT is self-evidently consistent.
The basic relationship between the parameters on both sides of the duality is 
$$g_{\rm string} = g_{\rm YM}^2, \quad \frac{R^4}{\ell_{\rm string}^4}  = g_{\rm YM}^2 N
\equiv \lambda $$
So at a fixed $N$, the weak coupling of the Yang-Mills side coincides with the weak string coupling in the type IIB string theory bulk.
When $N$ is allowed to scale to infinity as well, the 't Hooft coupling $\lambda\equiv g_{\rm YM}^2 N$ is what decides whether the loop diagrams are actually suppressed. 
You see that when $\lambda$ is  smaller (or much smaller) than one, then the Yang-Mills expansion is weakly coupled and the perturbative gauge-theory diagrams are guaranteed to approximate physics well (or very well). On the contrary, when $\lambda$ is greater (or much greater) than one, the AdS radius $R$ is greater (or much greater) than the string length which means that one may approximate the physics by string theory on a "mildly curved" background. 
In this limit, when the curvature radius is (much) longer than the string length, it is always possible to approximate low-energy physics of string theory by supergravity. In string theory, the SUGRA approximation means to neglect the $\alpha'$ stringy corrections. In the gauge-theoretical language, it means to focus on the planar limit for large $\lambda$ and neglect $1/N$ nonplanar corrections.
However, it's been demonstrated that all the "beyond supergravity" states you expect to see in the type IIB background appear on both sides of the AdS/CFT correspondence, including arbitrary excited strings) – this is particularly clear in the BMN/pp-wave limit (see also 1,000+ followups) – as well as various wrapped D-branes and, what is critical for the usefulness of the whole AdS/CFT framework, evaporating quantum black holes.
