EUD is a technique that lets triggers read and write StarCraft's own memory. It began as the death-counter condition: hand it an out-of-range value and it addresses memory that is not a death counter at all.
That unlocks a specific set of things: text drawn wherever you like in whatever colour, unit hit points and ranges changed at runtime, minimap drawing, key input. The HUD and skills you see in a modern map are essentially all EUD.
EUD runs inside the map. It is compiled into the map file rather than being an external program, so players install nothing.
EUD is version-sensitive: a patch can move the addresses. Tools like EUD Editor 3 track them per version for you, which is why you should not hardcode an address yourself.