Well, as near as I can figure out, it's King Ghidorah (aka Ghidrah, the Three-Headed Monster). The big guy has taken up residence on Planet X, a newly discovered planet in orbit around Jupiter. Two astronauts (one of whom is American actor Nick Adams), land there and meet the weird aliens who inhabit the planet. The aliens promise a cure to cancer if the Earth will help them rid their world of the menace of King Ghidorah. The help? Godzilla and Rodan who are captured on Earth and transported to Planet X to battle the three-headed monster.
The battle over, all three monsters are then returned to Earth where they proceed to attack various cities while under the control of the aliens (who, of course, turn out to be bad guys). There is much destruction and devastation before a way is found to free the monsters from the aliens' control, allowing them to fight the aliens and each other. The film ends with King Ghidorah flying off to who knows where and the fates of Godzilla and Rodan unknown. But don't worry. They're not dead. They both came back in subsequent giant monster films produced by Toho Studios.
INVASION OF ASTRO-MONSTER has fairly good special effects, model work and miniatures. The problem is that by this point in the series, Godzilla was already starting to be played for comic relief instead of the fearsome creature of mass destruction that was seen in the first two films, GOJIRA (GODZILLA) (1954) and GODZILLA RAIDS AGAIN (1955). By the mid-60s', the Godzilla films and their brethren were clearly being produced and marketed to kids and, if taken in that spirit, make for some fun, entertaining movies.
I watched the original Japanese version of this film in Japanese with English sub-titles. I find that far more preferable to watching a horribly dubbed film in English. It's a bit funny to hear Japanese coming from Nick Adams' mouth but it's a more authentic experience.
INVASION OF ASTRO-MONSTER is classic Saturday afternoon matinee material. It's bright colors, fast moving plot and three giant monsters guarantee a good time. I can't watch Japanese giant monster movies on a steady basis. About one of these babies a year is just enough for me but I do enjoy them.