BCNM stands for "Burning Circle Notorious Monster"
Scattered across the world are several "Burning Circles" that can be used to start boss-type fights. In some cases the circles are used for plot-driven fights, like the Rank 3 Dragon fights.
Many of the circles can also be used for Orb fights, which are most often referred to as BCNMs. You can trade Beastman Seals and Kindred Seals to a dealer in Port Jeuno for various orbs. When you trade one of these orbs to a circle you trigger a fight with a set of NMs. These fights generally cap your level and restrict the number of people that can enter the circle.
BCNMs also often have very good drops (Mannequin parts, expensive spells, expensive armor, etc) so sometimes groups will form to do BCNM fights for fun and profit.
Here's an
example of an Orb BCNM fight.
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Splaytree - Elvaan - Alexander
62-SMN / 50-RDM / 38-WHM / 28-BRD / 26 - DRG / 25-BLM
So you want a mannequin? 69.1 +2
{Goldsmithing
} "Curse your sudden but inevitable betrayal." ~Wash