Not sure about logograms (although I presume it's a currency used in Eureka), but NM stands for 'Notorious Monster', it's an old FFXI term. NMs in FFXIV are the 'elite' monsters that appear in many different content in the game, from FATEs (where you have to fight a single large monster), to the Hunt (which is technically the closest FFXIV has to what NM actually meant in FFXI), to special monsters that appear in content like Eureka and the Diadem. As I understand it, the Eureka NMs drop currency needed to getting special gear and upgrading the Stormblood relic weapon so they're highly sought after.