"The Fish" was Chris Squire's nickname. He earned it primarily due to his very long baths he would take, back when several members lived in the same flat and had to share the bathroom. Bill (Bruford) especially was not amused. He also happens to be a Pisces.
The subtitle for this song is "Schindleria Praematurus," which is an obscure, neotenic marine fish from the Pacific ocean. "Neotenic" means the adult fish exhibits no adult characteristics, only juvenile characteristics. The story is that Chris Squire had the melody and wanted to sing the name of a fish that had eight syllables, and dispatched a roadie (Maybe Michael Tait) to find one. The best he could find had nine, which is why the last syllable kind of trails off.
Squire's solo also incorporates much of the bass lines from the song "Tempus Fugit", which is a Latin phrase usually translated into English as "Time Flies".