User summary, details from Japanese Wikipedia:
The sequel to Bubble Bobble, with the subtitle "The Story of Bubble Bobble 2" although the gameplay is different, and the only real connection is that the main character is the same, although reverted to human child. The player can build small rainbows in front of him, and when jumping on them, they drop down and will defeat any enemy they touch. Players can also use the rainbows to climb up onto higher places, as the screen scrolls upwards.
The year after the release, the arcade "Rainbow Islands extra" was released, with adjusted difficulty and the order of the levels changed.
There are a number of remakes, such as "Party's Party" for the Wonderswan (2000), "New Rainbow Islands" for the DS (2005), "Hardy Gardy Adventure" for the PSP (2007) and Towering Adventure for WiiWare and Xbox Live Arcade (2009)."
Source: Wikipedia, "レインボーアイランド," available under the CC-BY-SA License.