What is it? It's a part of science which is going to matter to tort lawyers. In short, it's what we presently label as the control system which can alter the instructions programmed into our genes.
Why does it matter? Because alterations to the epigenome can alter the behavior of the genome, producing, for example, extreme changes (e.g. cancers; see this Nature article on melanomas) and more subtle changes (coloration; obesity). Of significance, the alterations are at least sometimes marked and observable at the cellular level. Imagine what these markers can do for toxic tort litigation and insurance coverage issues.
The changes wrought by epigenomic changes may be multi-generational. Examples ? Consider this December 2010 Nature article by Farooq Ahmed which describes in lay terms how starvation of parents appears to have altered genomes for their children. Or take a look at this Nature article by Anna Petherick on how the contents of breast milk in mammals differ throughout nursing. Or this Nature article on the emerging science of "nutrigenomics." For something much darker, note this Time article on research suggesting that the stresses induced by the Nazi have caused genetic changes in children of the parents who were tortured.
For more on epignetics in general, this Time article from late 2009 marked the first wide mapping of a part of the human epigenome by Salk Lab scientists, as reported here in Nature. Or, enjoy this January 2010 John Cloud article in Time, titled Why Your DNA Isn't Your Destiny. The article includes helpful illustrations.