Archonstorm is a nickname given to a situation where a player is on a level alongside a lot of Archons in a large group. What makes those kinds of groups significant, apart from Archons being rather formidable foes by themselves, is that when the number of Archons on the screen reaches a certain number, inversely proportional to the number of monsters you have genocided, they will continue to summon new Archons exponentially with Summon nasties, eventually filling the entire screen. This is most commonly seen with Plane of Fire, being an endgame level and thus eligible to summon Archons, and being a fully open level. It also occasionally happens on Air, and very rarely in other planes.
Archonstorms, and Archons in general, are the major reason why it is discouraged to perform excessive genociding of monsters.