The Time Devourer came from the game prior to Chrono Cross, which is Chrono Trigger. In CT, the main villians were the Lavos species. Even after their destruction, their Mammoth Machine still survived after the battle, and is still located in the year 2400AD, and is in the process of destroying all time and existence, hence it's name. Trapped inside is the young Princess Schala, who's life is in danger. In CC, at the end, you must destroy the Time Devourer without killing Schala as well by playing the "Song of Life," which is playing the element colors in order, then using the "Chrono Cross" element.
The Time Devourer is my favorite boss battle from the video game Chrono Cross.