Epigenetics is the study of how biochemical modifications or physical interaction of DNA/chromatin affect gene regulation in a cell, where such modifications/interactions are not related to changes in the underlying DNA sequence.
At the DNA level, methylation of CpG dinucleotides (often located near gene promoters) can be detected by first converting unmethylated cytosines into uracil using bisulfite, which allows methylated and unmethylated cytosines to be distinguished.
At the chromatin level, modifications of the tails of histone proteins (e.g. methylation, acetylation) can be mapped by 'immunoprecipitation', where chromatin and proteins are chemically cross-linked reversibly. The genomic DNA associated with the modification/protein of interest is then 'pulled-down' (precipitated) with specific antibodies raised against the modification/protein. After precipitation, the cross-linking is reversed to release the genomic DNA for further analysis.