Real name: Prince Rogers Nelson

Gemini is not a Batman character, but Prince's own alter-ego, and was not, nor ever intended to be, part of the Batman or DC Comics canon.
Gemini was an alter-ego created by Prince for his work on the Batman project, and was depicted with one side representing the Joker (evil), and the other Batman (good). Gemini appeared in the music videos for Partyman and Batdance, and on the album is credited with vocals for the latter.
Gemini is Prince's astrological sign, and was used as a reference to the duality bewteen sex and religion throughout his music.