I would recommend the article 2D Turbulence from Boffetta and Ecke published in the Annual Review of Fluid Mechanics in 2012. It gives equations and statistical informations in spectral space with known scalings on energy spectra and third-order structure function. Then details about methods and examples on different approaches are mentioned and finally, results are shown using both numerical and experimental data. A clear summary and some future issues are addressed at the end. In the introduction, they mention others reviews which could help you evaluate the progression in the subject. I especially recommend :
- Two-dimensional turbulence by Kraichnan and Montgomery in 1980. It displays analytical developments and numerical results. Very interesting but maybe not the best paper to begin with, may be hard to get the idea for someone without a good mathematical background.
- Two-dimensional turbulence: a review of some recent experiments by Kellay and Goldburg in 2002. As it says, their review is based on experimental results with both decaying and forced turbulence as well as mixing of a passive scalar. They recall the basics of 2D turbulence in a small introduction.
A forth source could be the book of Lesieur, Turbulence in Fluids. The eighth chapter is dedicated to two-dimensional turbulence. You will find everything, statistical tools, enstrophy, inverse cascade, transfer, etc. He also mentioned how to study 2D turbulence with a spectral model using an EDQNM closure.
Those four references will certainly give you a complete view on the 2D turbulence fundamentals. Feel free to complete if I missed a major article.