This article is from the Anime FAQ, by Steve Pearl with numerous contributions by others.
DAICON is a yearly SF convention held in Osaka, Japan. It is called DAI-CON because the kanji for Osaka can be read as "dai". (A "daicon" is also the name for a Humungous white radish, but that's another story.)
"DAICON III" and "DAICON IV" were two animated shorts shown at the opening ceremonies of the 15th (1981) and 17th (1983) DAICONs. They feature the convention mascot (the Daicon Bunny, strongest Playboy Bunny in the universe) encountering everybody from the Space Cruiser Yamato to Ming the Merciless.
As an interesting note, the people who made DAICON IV later went on to create the GAINAX (WINGS OF HONNEAMISE, NADIA OF THE MYSTERIOUS SEAS) animation studio.