Before we go any further, I'll answer the first question I am always asked: no, it is not that thing at renaissance fairs where people dress up in costumes, hold royal courts, and often beat one another up with foam tubing. That is most likely the Society for Creative Anachronism or some other role playing, primarily creative, affiliation. It is also not theatrical fighting or a fencing sport. All three of these may or may not draw from historical materials or origins but do not observe the actual method of practical survival with historical weapons.
HEMA is exactly that. The term refers to the documented methods of historical European combat. It is used as a system for reconstructing medieval combat according to its original function through study and practice. You can read more about it here at the HEMA website.