Hierarchical Cities: After the Black Beast was defeated, the world was covered in deadly amounts of Seithr. Many of the remaining people escaped to the tops of mountains and created the Hierarchical Cities. The NOL offices are at the top, then the wealthy citizens, and then the poorer citizens toward the bottom. All the Hierarchical Cities are built on top of a Cauldron. The cities are modelled after Ishana.
Azure Grimoire: A High Density mass of concentrated Seithr. Not really a traditional Grimoire, the Azure Grimoire allows someone to directly access part of (or maybe a copy of) the Azure. This makes it much more powerful than something that just uses Seithr. Terumi created the Azure Grimoires. Terumi owns one and Ragna owns the other. The Azure Grimoire is said to be like a Cauldron. It can gather souls and can be used to destroy Cauldrons.
Duodecim: Twelve families that quickly adopted Ars Magus. These are the big families of the NOL and many of the highest ranking NOL members come from these families. Kagura, Tsubaki, Jin, and Mai all belong to Duodecim families.
Orbis Librarius Norma / OLN / NOL / Novus Orbis Librarium (世界虚空情報統制機構 Sekai Kokū Jōhō Tōsei Kikō: World Void Information Regulation Organization) / Librarium (統制機構 Tōsei Kikō: Regulation Organization): The NOL was originally established during the Dark War to help create Ars Magus and the Grimoires. After the war, it dedicated itself to controlling access to the Grimoires and started being known as the Library. It militarized itself in order to protect the Grimoires, then rapidly used its forces to take more and more control of the world. It pretty much runs what's left of humanity at this point. The NOL is ruled by the Imperator (帝 Mikado: Emperor).
Sector Seven (第七機関 Dai-nana Kikan: Seventh Institution):
It's not clear when Sector Seven was established, but the Chronophantasm library places it as existing before the Dark War and says Terumi founded Sector Seven.
The modern Sector Seven wishes to restore the world of the past, not relying on Ars Magus or Seithr. They focus on exclusively using science over magic. They search for ways to nullify Grimoires. The executive officers of Sector Seven are called the Committee. The NOL is said to be their main adversary and polar opposite, but at their absolute highest levels, both Sector Seven and NOL are secretly controlled by the same people.
Kokonoe is a Chief Scientist of Sector Seven, but she in no way controls Sector Seven. She had the Sector Seven facility built on top of the Cauldron in the First District.
Artificial Causality Weapon / Artificial Causality Phenomenon Weapon (模倣事象兵器 Mohō Jishō Heiki: Imitation Phenomenon Weapon): A non-magical Nox Nyctores that operates via atomic power instead of via Seithr. They are usually less powerful than Nox Nyctores, but can continue to work when Seithr is nullified. Kokonoe developed these after studying Nirvana. Iron Tager, Minerva, modern Nirvana, Ignis, and Lambda all use Artificial Phenomenon Weapons. The IDEA Engine is described as an upgraded Artificial Causality Weapon, but the exact details aren't provided. Relius created Ignis, which is also an Artificual Causality Weapon (but he uses the term Detonator.)
Power of Order: I don't totally understand this. This is the power some characters have that allow them to destroy Phenomenon through sheer force of will. Multiple characters can use this, but Jin is the most mentioned user. It seems to be related to the desire to either keep the world from changing too much or stop the world from being destroyed.
Relius and Terumi's Partnership: Relius and Terumi have partnered with each other for centuries. Their single goal is to create a Kusanagi and then destroy the Master Unit. The Black Beast, NOL, Imperator, and Sector Seven are all tools they use for their goal. Afterwards, Relius wishes to recreate existence to his liking.