Why is Sachdev-Ye-Kitaev (SYK) Model model important? In the past one or two years, there are a lot of papers about the Sachdev-Ye-Kitaev Model (SYK) model, which I think is an example of $\mathrm{AdS}_2/\mathrm{CFT}_1$ correspondence. Why is this model important?
 A: The other answers already pointed out very important properties, but there is a further aspect related to black hole physics. Namely, $AdS_2/CFT_1$ is the relevant holographic description of four dimensional extremal black holes, for instance the near horizon limit of an extremal Reissner–Nordström is $AdS_2 \times S^2$.
Holographic techniques allowed the comparison of five dimensional black holes microstates between small and large string coupling constant regime. The same technology is not available for 4d black holes, and one must use other tools like the supersymmetric quantum mechanics on the worldvolume of the intersecting branes forming the black hole or string theory scattering amplitudes. 
A: People hope that it may be an example of AdS/CFT correspondence that can be completely understood.
AdS/CFT correspondence itself has been an incredibly important idea in the hep-th community over the past almost twenty years.  Yet it remains a conjecture.  In the typical situation, quantities computed on one side of the duality are hard to check on the other.  One is computing in a weakly coupled field theory to learn about some ill defined quantum gravity or string theory.  Alternatively, one is computing in classical gravity to learn about some strongly interacting field theory where the standard tool box is not particularly useful.
The original hope was that SYK (which is effectively a quantum mechanical model)  might have a classical dilaton-gravity dual description in an AdS$_2$ background.  That hope seems to have faded among other reasons because the spectrum of operator dimensions does not seem to match (see e.g. p 52 of this paper).  Yet, there still might be a "quantum gravity" dual, for example a string theory in AdS$_2$.  String theories in certain special backgrounds have been straightforwardly analyzed.  
A: SYK model provides us with the simplest example of holography which is much easier to study than canonical $AdS_5 \times S^5$ case due to much lower dimensionality. It was the initial motivation for Kitaev to study this model. Here is a set of 2 lectures in which he briefly discusses it.
Because of its simplicity, it is easy to consider the thermal and chaotic behavior of this theory and its gravity dual. Look at the following papers for the details:
Maldacena, Stanford "Comments on the Sachdev-Ye-Kitaev Model". It describes the correspondence in details.
Maldacena, Stanford, Yang "Conformal Symmetry and its Breaking in Two Dimensional Nearly Anti-de-Sitter Space". This paper describes the gravity side of the correspondence. In particular, modified gravity on the N(early)AdS space on which the bulk theory must live, because usual GR is trivial in 2D.
Shenker, Stanford "Stringy Effects in Scrambling". Here the stringy effects which must be taken into account in addition to field-theoretical gravity in the bulk are discussed.
