EUD's are Extended Unit Deaths. Basically, before it was patched, you could modify starcraft's memory by setting the deaths of an extended unit to a specific number. Though this offered unimaginably more options in mapping, it also was a security risk, so blizz patched it.
Even though you can't set deaths with EUDS anymore, you can still use them for conditions. It is mainly used to detect the health or energy of a pre-placed unit, screen position.. random stuff. Here's the recent thread that mentions it.
http://www.staredit.net/index.php?showtopic=36621