Kahler spaces are just certain singular spaces equipped with a Kahler metric in appropriate sense. I first came across it Demaily-Paun's classical paper Numercical Characterization of the Kahler cone of a compact Kahler manifold. However it seems to refer the reader to the relevant background material in another paper of Demaily which is in French. I wonder whether there is any reference that have some background materials on Kahler spaces: definition, fundamental properties etc.
There is a discussion of this topic in the recent book Cycles analytiques complexes II : l'espace des cycles by Barlet and Magnússon, chapter XII.3. An English translation (to be published by Springer) is forthcoming.
You will also find a short discussion in the book Degenerate Complex Monge–Ampère Equations by Guedj and Zeriahi, chapter 16.3.
For a reference in English, you could take a look at this paper of Varouchas, which contains a definition of Kähler spaces and relatied concepts (e.g. Kähler morphisms) and some of their fundamental properties.