HEMA, or Historical European Martial Arts, is a collection of independent martial systems that trace their roots back to medieval Europe - specifically, martial systems that originate in Europe to which there is no living lineage in the modern day, but surviving written records (from instruction manuals to firsthand accounts) do exist for us to base our research on.
Modern HEMA involves the reconstruction of these systems through research of surviving manuscripts, application of techniques and "stress testing", and some clever interpretation and experimentation.
The HEMA community is a group of dedicated martial artists and academics who faithfully attempt to recreate the martial practices of Europe from antiquity through the 20th century. Unlike some more common martial arts, this involves a greater focus on weapons and their usage, including longsword, spear, rapier, messer, saber, sword and buckler, bayonet, spadroon, and more.
HEMA also focuses on the training methods utilized historically for these techniques, including training weapons like the single-stick and "federschwert", and exercise routines linked closely to the practice of historical swordsmanship.
HEMA has both an academic side and a competitive side - our club engages in both. There are local and national HEMA tournaments where you can test your skill against other fighters in a competitive environment, there are tatami cutting competitions, and there are a host of ways to get involved with the research and academic side of HEMA even as a beginner! HEMA truly has something for everyone.
TL:DR; it's old-fashioned sword fighting with modern safety equipment and probably some reading. (No, we aren't a LARP, we're a martial art, but we'd be happy to recommend some great LARPs in the area if you're interested in that sort of thing!)