arena - The inside of the white box in which the player is allowed to move.
bullet - Everything in a wave that can collide with you. Flowey's pellets would be referred to as bullets, but so would Papyrus' bones, anything Woshua can shoot (of any colour), even the dancing Migosp.
encounter text - The text that shows up before you've selected FIGHT/ACT/ITEM/MERCY.
monster dialogue - Text from monsters in an encounter, often seen before attacking. Can also be multiple dialogue boxes for special encounters.
dialog - A user interface component that contains text. For example, the battle dialog window. The distinction between "dialog" and "dialogue" is that "dialog" refers to interface windows containing text, and "dialogue" refers to the speech content of monsters.
wave - A single attack behaviour (or attack "wave", to say), measured from when you start defending until when it stops. Vegetoid's bouncing vegetables attack would count as a wave. Papyrus' special Cool Dude attack would also count as a wave. Unitale works with "wave scripts" for attacks; you can use multiple wave scripts at the same time for when you have various monsters.
overworld - A CYF feature that is external to battles. It allows
the user to explore maps, like in Undertale. The overworld can contain event objects, too.
You can find a tutorial
on maps in How to create a map and a tutorial on events is in How
to create an event.
modDev - This is a feature exclusive to CYF. In CYF, you can
access some extra options from the Mod Selection screen. These mainly include the options to wipe
RealGlobals and AlMightyGlobals, as well as toggle
safe mode and retrocompatibilty mode.
* In CYF v0.6.2.1 and beyond, the modDev screen is located in an "options menu" accessible by clicking "Options"
from within the Mod Selection screen.
retrocompatibilty mode - This CYF feature is designed in such a way where, if it's enabled, functions from 0.2.1a that were changed in CYF will function exactly as they did in 0.2.1a.
safe mode - This is a CYF-mode that removes swear words found in CYF. Mod authors can also use this to do the same in their mods, maybe even also using this to remove blood or mature content.