Does AdS/CFT correspondence take place only near black holes? Is it related to black holes or is AdS/CFT a separate thing itself?
 A: I’m having trouble understanding your question. I think you’re asking whether AdS/CFT correspondence is inherently rooted in black hole physics or not. Or, perhaps, you're asking one has to consider a black hole system for applying the AdS/CFT. Is that correct? I try to express some concepts which may be your answer.
AdS/CFT is a conjectured relationship between two kinds of physical theories. On one side are anti-de Sitter spaces (AdS) which are solutions of gravitational theories (including general relativity, supergravity, string theory and M-theory). On the other side of the correspondence are gauge field theories which are quantum field theories including conformal field theories (CFT) such as  N=4 Supersymmetric Yang–Mills theory. For example, by studying two different limits of the D3-brane system one can perfectly understand the $AdS_5$/$CFT_4$ version of the conjecture. Always, the two different limits (descriptions) of the brane are complementary to each other, but the mysterious AdS/CFT dictionary relates those different descriptions.
Concisely, AdS/CFT claims that a gravitational theory in a $d+1$-dimensional AdS background is related to a gauge field theory (perhaps with conformal symmetry) in $d$-dimensions. AdS black holes are solutions of those gravitational theories, so the thermal partition function of a specific AdS black hole will be equivalent to the partition function of a gauge field theory at the boundary. By "boundary", I mean at asymptotic region $r \to \infty$ (i.e., very far from black hole). So, naively speaking, the boundary of AdS-gravity (the asymptotic behavior of AdS black hole, i.e., $r \to \infty$) is equivalent to a specific gauge field theory located at the boundary.
Black hole are thermal systems and the black hole temperature at the boundary can be obtained by the Hawking radiation formula. This temperature is interpreted as the temperature of the gauge field theory. So, using AdS black hole physics, we could have a gauge field theory at finite temperature at the boundary. Of course, one can consider an AdS space without having any black hole and then the resulting gauge field theory at the boundary would be at zero temperature (in fact, quantum field theories in textbooks are in zero temperature and QFT at finite temperature is a little more advanced subject). Generally, when we want to discuss about the finite temperature gauge field theory in AdS/CFT correspondence, we have to use the finite temperature solution of the D-brane system which always is a black hole (this black hole solution is obtained by taking a certain limit of that D-brane system). For these reasons, you see that the AdS/CFT correspondence always appears with the notion of the black hole.
If you are interested in the subject of AdS/CFT, I highly recommend this book (excellent for beginners).
A: While the AdS/CFT correspondence is related to black holes and a tool for studying them, it is a much broader framework. Generally speaking, it concerns a correspondence between a quantum field theory defined on anti de Sitter space and conformal field theory.
Anti de Sitter space is the maximally symmetric, Lorentzian manifold with constant negative scalar curvature, which is to say the Ricci scalar $R = g_{ab}R^{ab} <0 $.
On the other side of the correspondence is a conformal field theory, which is a quantum field theory that has conformal symmetry. This is generated by,

*

*Translations $x \mapsto x^\mu + a^\mu$

*Dilations $x^\mu \mapsto \lambda x^\mu$

*Rotations $x^\mu \mapsto \Lambda^\mu_\nu x^\nu$

*Special conformal transformations which are a combination of an inversion, translation and another inversion, where by inversion we mean $x^\mu \mapsto x^\mu/|x|^2$.

Conformal field theories have a more rigorous formulation in terms of vertex algebras and vertex operator algebras, which are associative algebraic structures you can think of as  a vector space with an infinite family of left-multiplications, indexed by an indeterminate.
