The histone code is a hypothesis that the transcription of genetic information encoded in DNA is in part regulated by chemical modifications to histone proteins, primarily on their unstructured ends.  Together with similar modifications such as DNA methylation it is part of the epigenetic code.
Cumulative evidence suggests that such code is written by specific enzymes which can (for example) methylate or acetylate DNA (writers), removed by other enzymes having demethylase or deacetylase activity (erasers), and finally readily identified by proteins (readers) that are recruited to such histone modifications and bind via specific domains, e.g., bromodomain, chromodomain. These triple action of writing, reading and erasing establish the favorable local environment for transcriptional regulation, DNA-damage repair, etc.
The critical concept of the histone code hypothesis is that the histone modifications serve to recruit other proteins by specific recognition of the modified histone via protein domains specialized for such purposes, rather than through simply stabilizing or destabilizing the interaction between histone and the underlying DNA. These recruited proteins then act to alter chromatin structure actively or to promote transcription.
A very basic summary of the histone code for gene expression status is given below (histone nomenclature is described here):