The Showa period is the era in Japan ruled by Emporer Hirohito, or the "Showa Emporer" posthumously. Hirohito's reign lasted from 1945-1989.
The Heisei period is the era in Japan ruled by Hirohito's son, Akihito. It started the day after Hirohito's death in 1989. The Heisei period is still going on in Japan since Akihito is still alive.
Because all of the original Godzilla movies from 1954-1975 take place in the Showa period of Japan it's known by fans as "The Showa Series". Although The Return of Godzilla/Godzilla 1985 also takes place in the Showa Period of Japan, all of its sequels were released from 1989-1995 during the Heisei Period of Japan. So even though The Return of Godzilla/Godzilla 1985 is technically a Showa Period film, it's a Heisei Series Godzilla movie due to the continuity of the sequels which take place during the Heisei Period. (It simply makes sense to lump the film with its sequels.)
The Millennium Series is called a different name to separate it from the Heisei series, even the though the Heisei Period is still going on in Japan. I've heard that Japan doesn't seperate the Godzilla series with the Showa and Heisei names as often as Westerners do though. Apparently in Japan the Showa Series is called "The Original Series", the Heisei Series is called "The VS Series" and the Millennium Series is the same.
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