Majesty (マジェスティ Majesutei): A gem-like object that is found after defeating CIMA. They can be used to synthesize items, and react differently to every person; that's why some characters can make items for different costs than others. Four types exist: Rectangles (the most common, found on easy enemies), Diamonds (found on flying CIMA), Spheres (found on strong CIMA) and Stars (rare and seem to be found on varying types of CIMA). Boss class monsters don't drop these, nor do the transformed builders.
Gate (ゲート Geeto): A portal that connects the human world to the Gate World. It can be opened by CIMA builders.
Gate World (ゲート ワールド Geeto Waarudo): A term coined for the world that CIMA live in.
Builder CIMA (ソルジャー クラス Sorujya Kurasu): Humanoid in shape, these CIMA are responsible for creating the dungeons used to trap humans. Some have human skin tones, while others have strange colors such as purple. It also appears that they have pointed ears (evident from the three in-game whose ears aren't hidden by hair), but whether or not that's true for all builders is yet to be seen. In the Japanese version, while they were usually referred to as "High Class CIMA" (ハイ クラス の CIMA), Ark and Ivy make a direct reference to "Soldier Class" after the second boss fight, which makes me think that "Soldier" is the real term while "High" is referring to their ability level.
Yes, the word "builder" is generally written in lower case, barring one instance in a Talk session in Beginning World, despite the fact that its use as a title should make it a proper noun. Though I'm writing it as seen in-game, I take it as someone in the proofreading department at Natsume missed that one (especially considering both Mommy and Daddy aren't capitalized when Halley and Emmy directly address their parents).
Crimson Nine (クリムゾン・ナイン Kurimuzon Nain): Nine most powerful builder CIMA in existence. Main antagonist Pike Nighttrap is one of them.
Dungeon (ダンジョン Danjyon): Where CIMA trap humans after bringing them to the Gate World. Since CIMA devour human hope and life force, there will always be a means of escape from one, it's getting there that's the hard part.
Dungeon Key (カギ Kagi, lit. Key): Needed to open the door to the builder's lair.
Teleport Trap (ワープ トラップ Waapu Torappu): aka "CIMA's Trap of Choice." Used to teleport people to other areas of the dungeons, they manage to split the entire group in two no less than six times throughout the game. They were called "Warp Traps" in the Japanese version.
Gate Guardians (ゲート ガディアン Geeto Gadeian): Humans who fight CIMA in order to protect others. Their motto is "Always Protect Everybody."
Six Stars (シックス スターズ Shikkusu Sutaazu): The six most powerful Gate Guardians, antithesis of the Crimson Nine. Ark and Ivy's mentor, Jester R, is one of them.
Trust: Can be gained by fighting CIMA near the in-party characters and successfully preventing them from being hurt. Inversely, if they are hurt in the dungeons (be it by enemies or traps), Trust can be lost. Both gains and losses can also occur in cutscenes. Characters that Trust you (represented by a green heart) will make items for you while those that don't (represented by a red heart) will not.
Singularity (特異点 Tokuiden, lit. Singular Point): To quote Pike himself, "The point at which the mass of everything reaches infinity! The point that swallows everything and leaves nothing behind!" Most fans seem to liken it to a Black Hole in theory. Both CIMA and the Gate Guardians want to open it, most likely to annihilate the other. It requires both the Rare Majesties as well as the correct person to resonate with them.
Rare Majesties (レア・マジェスティ Rea Majesutei): Powerful Majesties capable of opening the Singularity. However, the Plug is needed to resonate with them.
Plug (プラグ Puragu): A person who resonates with the Rare Majesties to open the Singularity. It isn't known if different Rare Majesties have different Plugs or not, but the two in Pike's possession resonated with young Halley and Emmy H.
Base (ベース Baasu): This has yet to be explained in the context of the game. Some have theorized that it is what can stop the Singularity, while others think it is used to keep it in control and only attack certain targets. Most likely, like the Plug, it is a person.
Blue Creek (ブルークリーク Buruu Kuriiku): The train you're riding in. Claude is its conductor.