The AdS/CFT correspondence is one of the largest areas of research in string theory. AdS/CFT stands for Anti-de Sitter/Conformal Field Theory, an expression that’s not particularly elucidating.
AdS/CFT is a particular, and deeply surprising, example of a duality. It relates two very different theories and at first sight seems obviously wrong. It states that there is a duality between theories of gravity in five dimensions and quantum field theories (QFTs) in four dimensions. This correspondence was first formulated by Juan Maldacena in 1997, and is generally thought to be the single most important result in string theory in the last twenty years.
The original example of AdS/CFT linked two very special theories. The gravitational side involved a particular extension of gravity (type IIB supergravity) on a particular geometry (5-dimensional Anti-de-Sitter space). The QFT was the unique theory with the largest possible amount of supersymmetry. There’s a specific dictionary that translates between the theories.
This relationship has no formal mathematical proof. However a very large number of checks have been performed. These checks involve two calculations, using different techniques and methods, of quantities related by the dictionary. Continual agreement of these calculations constitutes strong evidence for the correspondence.
The first example has by now been extended to many other cases, and AdS/CFT is more generally referred to as the gauge-gravity correspondence. Formally this is the statement that gravitational theories in (N+1) dimensions can be entirely and completely equivalent to non-gravitational quantum field theories in N dimensions.
The AdS/CFT correspondence has a very useful property. When the gravitational theory is hard to solve, the QFT is easy to solve, and vice-versa! This opens the door to previously intractable problems in QFT through simple calculations in gravity theories.
Moreover AdS/CFT allows a conceptual reworking of the classic problems of general relativity. Indeed if general relativity can be equivalent to a QFT, then neither one is deeper than the other. Finally physicists can use it to develop new intuitions for both QFT and general relativity.